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THE DEVELOPMENT OF A MULTICHANNEL ATOMIC MAGNETOMETER ARRAY
FOR FETAL MAGNETOCARDIOGRAPHY
BY
ROBERT WYLLIE IV
A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(PHYSICS)
AT THE
UNIVERSITY OF WISCONSIN–MADISON
2012
i
ACKNOWLEDGMENTS
First, I would like to thank Thad Walker for making my experience along the road to this thesis
a completely positive one. His boundless enthusiasm and curiosity for his work will always serve
as an example for me, and he has been gracious with his praise and generous with credit for what
we have done. When potential students ask me how I like working with him, it’s a relief to be
able to tell them how unreservedly great that experience has been. Any future success I have as a
physicist, I owe in large part to him.
I should also mention my undergraduate institution, Denison University. In particular, the truly
extraordinary commitment of the physics department to helping their students flourish. I remember
several moments that still feel significant: Kim Coplin, with just a touch of exasperation, telling
me she didn’t know the answer to my question, before I asked it; Steve Doty telling me the two
ways to get rid of moss are starving it of sunlight and scrubbing it vigorously (the latter is the only
one that really works); Dan Homan imagining what Debye must have thought in the moment when
he learned his theory led to such an accurate description of specific heat in conductors, and many
others. I thank everyone there for helping prepare me for my experiences afterward.
Ron Wakai and his biomagnetism lab have been very helpful. They have greatly helped with
both the practicalities of working with physicists who know precious little about the signal they
are trying to detect, to more detailed technical assistance and guidance in the design and use of our
device.
My peers have done a great deal give me the experience I’ve had here. In particular, I’d like to
thank Nick Proite and Brian Lancor for many hours of discussion, helping me hone my thoughts
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about physics and everything else. I would also like to thank Matt Kauer and Greg Smetana, who
have contributed a great deal to the work in this thesis, as well as everyone else who has come
through Thad’s lab during my time here.
Finally, I owe an un-repayable debt of gratitude to my family. I thank Bill and Sarah Prueter for
treating me always with kindness and generosity. To my parents, Robert and Elaine, I am grateful
for unwavering love and support. Finally, I thank my loving wife and partner, Betsy. There’s no
telling where I would be without her influence, but surely not writing this. To her, I owe everything.
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TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Atomic magnetometer renaissance . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 SERF Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Historical optical magnetometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Traditional atomic magnetometry . . . . . . . . . . . . . . . . . . . . . . 72.1.2 The SERF regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 SERF magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Spin-relaxation from binary collisions and probe absorption . . . . . . . . 182.2.3 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Larmor precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.5 Equation of motion and steady-state response . . . . . . . . . . . . . . . . 232.2.6 Response to dynamic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.7 Sensitivity and Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.8 Probe Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Warts and imperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Spatial dependence of laser parameters . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Transverse spatial dependence of a Gaussian beam . . . . . . . . . . . . . 382.4.2 Longitudinal dependence of a weak probe . . . . . . . . . . . . . . . . . . 39
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Page
2.4.3 Longitudinal dependence of the pump . . . . . . . . . . . . . . . . . . . . 402.5 Light shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Cell, heatsink, and heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Optics and mounts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.1 Heaters and thermistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3.2 Coils and coil drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3.3 Data acquisition system . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Magnetometer array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Data Collection and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 Setting up magnetometers to operate in DC mode . . . . . . . . . . . . . . . . . . 674.2 Residual field nulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3 Characterization procedure using DAQ program . . . . . . . . . . . . . . . . . . . 70
4.3.1 Coil driving circuit calibration . . . . . . . . . . . . . . . . . . . . . . . . 704.3.2 Magnetometer calibration and initial noise measurement . . . . . . . . . . 714.3.3 Additional noise measurements (Probe and electronic) . . . . . . . . . . . 724.3.4 Chirp response measurement . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Signal measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Noise analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 Magnetic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1.1 Johnson noise from nearby conductors . . . . . . . . . . . . . . . . . . . . 855.1.2 Magnetic noise from nearby electronic circuits . . . . . . . . . . . . . . . 865.1.3 Noise from sources outside the MSR . . . . . . . . . . . . . . . . . . . . . 86
5.2 Fundamental quantum noise sources . . . . . . . . . . . . . . . . . . . . . . . . . 875.2.1 Spin-projection noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Photon shot noise (basic) . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.3 Experimental measurement of fundamental noise . . . . . . . . . . . . . . 905.2.4 Averaged parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2.5 Spatially dependent magnetometer response . . . . . . . . . . . . . . . . . 92
5.3 Technical noise sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.1 Signal aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.3.2 Probe polarization rotation in optical fiber . . . . . . . . . . . . . . . . . . 100
v
AppendixPage
5.3.3 Probe frequency and intensity noise . . . . . . . . . . . . . . . . . . . . . 1025.3.4 Pump laser fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.3.5 Laser Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6 Extensions and different operating modes . . . . . . . . . . . . . . . . . . . . . . . 107
6.1 Non-parametric field modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.1.1 Diffusion mode pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.1.2 Feedback mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Parametric field modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3 Z-Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4 Transverse pumping SERF magnetometer . . . . . . . . . . . . . . . . . . . . . . 129
7 Biomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.1 MCG characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387.2 Signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2.1 Single-channel processing techniques . . . . . . . . . . . . . . . . . . . . 1447.2.2 Multiple-channel processing techniques . . . . . . . . . . . . . . . . . . . 1457.2.3 Advanced nonlinear noise reduction . . . . . . . . . . . . . . . . . . . . . 1477.2.4 Comparison of SERF and SQUIDs for noise processing . . . . . . . . . . 149
7.3 MCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.4 Phantom measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.5 Fetal MCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
APPENDICES
Appendix A: Dipole electric polarizability . . . . . . . . . . . . . . . . . . . . . . . . 172Appendix B: Operating mode parameters . . . . . . . . . . . . . . . . . . . . . . . . 177Appendix C: Cell characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 180Appendix D: Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Appendix E: Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Appendix F: Fetal MCG report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Appendix G: Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Appendix H: Magnetometer parts list . . . . . . . . . . . . . . . . . . . . . . . . . . 206
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LIST OF TABLES
Table Page
3.1 Design requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Pumping parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1 Calibration frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Noise types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1 Noise budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1 Noise implications for signal processing . . . . . . . . . . . . . . . . . . . . . . . . . 143
AppendixTable
C.1 Nominal cell properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D.1 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
E.1 Heater controller parts list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
E.2 Coil control box features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.1 Fetal MCG report average measurements . . . . . . . . . . . . . . . . . . . . . . . . 195
G.1 Coil calibration: transversely pumped magnetometer . . . . . . . . . . . . . . . . . . 205
H.1 Magnetometer parts list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
H.1 Magnetometer parts list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
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LIST OF FIGURES
Figure Page
2.1 Mx magnetometer setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Spin-exchange collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 The SERF regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Slowing down factor and the spin-temperature distribution . . . . . . . . . . . . . . . 13
2.5 Relaxation rates from spin-exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Magnetometer pump/probe geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Spin relaxation due to collisions with atoms and walls . . . . . . . . . . . . . . . . . 22
2.9 Sensitivity and polarization vs. pumping rate . . . . . . . . . . . . . . . . . . . . . . 24
2.10 Magnetometer vector sensitivity-adjusting Ω⊥ and R . . . . . . . . . . . . . . . . . . 26
2.11 Magnetometer vector sensitivity-R=1 . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.12 Magnetometer swept field response . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.13 Magnetometer frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.14 Vector response to AC fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.15 Frequency response with actual relaxation rates . . . . . . . . . . . . . . . . . . . . . 34
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Figure Page
2.16 Probe Faraday rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.17 Diffusion length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.18 Spatial dependence of R and Pz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.19 Pump laser light shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Single magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Windows and mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Optical elements and the optics tube . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Electrical heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 H-Bridge schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Data acquisition flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.7 Swept field fit and ambient MSR gradients . . . . . . . . . . . . . . . . . . . . . . . 63
3.8 Magnetometer array schematic and photograph . . . . . . . . . . . . . . . . . . . . . 65
4.1 Amplitude and phase calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Noise measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Chirp signal calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Effect of probe intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Artist’s rendition of magnetometer position for a fMCG measurement . . . . . . . . . 82
5.1 Field from distance objects outside MSR . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Reference values for the photon shot noise given the magnetometer performance . . . 91
5.3 Spatially dependent modeling of the magnetometer, without any bias fields . . . . . . 95
5.4 Spatially dependent magnetometer performance, including ΩLS . . . . . . . . . . . . 96
5.5 Spatially dependent magnetometer performance, including ΩLS and an offset field Ω0 . 98
ix
AppendixFigure Page
5.6 Fiber polarization noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.7 Probe frequency and intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 δR vs. frequency fluctuations and δPx for δR . . . . . . . . . . . . . . . . . . . . . . 105
6.1 Diffusion pumping radial Pz and radial sensitivity . . . . . . . . . . . . . . . . . . . . 110
6.2 Diffusion pumping model through the cell . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Diffusion pumping noise measurement . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4 Feedback loop block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5 Feedback coil for feedback gradiometer . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.6 Feedback noise measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.7 Feedback performance for real time noise suppression of fMCG data . . . . . . . . . 121
6.8 Bessel functions in Fourier expansion of z-mode sensitivity . . . . . . . . . . . . . . . 126
6.9 z-mode noise performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.10 Pump-probe-large DC field geometry for the transversely pumped SERF magnetometer 131
6.11 The DC term of the response for Px and Py for the transversely pumped SERF mag-netometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.12 Optimum pulse shape and a cartoon of the polarization dynamics for the transverselypumped SERF magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.13 Simulation for non-ideal pulse area . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.14 Initial transversely pumped resonance measurements . . . . . . . . . . . . . . . . . . 137
7.1 Simulated MCG and fMCG signal and bandwidth . . . . . . . . . . . . . . . . . . . . 140
7.2 STFT of the simulated MCG signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3 Nonlinear signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
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Figure Page
7.4 Adult MCG measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.5 Removal of periodic interference from an MCG signal . . . . . . . . . . . . . . . . . 153
7.6 Phantom MCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.7 Raw fMCG measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.8 Calibrated fMCG measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.9 Fetal MCG analyzed using PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
AppendixFigure
B.1 DC-mode optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.2 Diffusion pumped optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
C.1 He diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.2 Uncalibrated By sweep showing typical channel-to-channel variation . . . . . . . . . 181
E.1 Heater H-bridge controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
E.2 Nominal thermistor calibration for magnetometer temperature measurement . . . . . . 190
E.3 Coil control box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
E.4 Noise measurements for coil driving circuit settings . . . . . . . . . . . . . . . . . . . 194
F.1 Average SERF fMCG measurement with relevant feature intervals . . . . . . . . . . . 196
F.2 Average SERF fMCG measurement from another time segment . . . . . . . . . . . . 197
F.3 Average SQUID fMCG measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 198
G.1 Calibration for the individual coils wrapped around each magnetometer and field uni-formity simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
G.2 Field uniformity for the large coil used in feedback mode . . . . . . . . . . . . . . . . 202
xi
Figure Page
G.3 Transversely pumped magnetometer coil calibration and simulation . . . . . . . . . . 204
xii
ABSTRACT
Biomagnetic signals can provide important information about electrical processes in the hu-
man body. Because of the small signal sizes, magnetic detection is generally used where other
detection methods are incomplete or insufficiently sensitive. One important example is fetal mag-
netocardiography (fMCG), where the detection of magnetic signals is currently the only available
technique for certain clinical applications, such as the detection of cardiac arrhythmia. Until now,
magnetometers based on superconducting quantum interference devices (SQUIDs), which can op-
erate at sensitivities down to 1 fT/√
Hz have been the only option. The low Tc superconductors
and associated cryogenics required for the most sensitive devices has led to interest in alternative
technologies. In the last decade, atomic magnetometers operating in the spin-exchange relaxation-
free (SERF) regime have demonstrated a higher sensitivity than SQUIDs while operating near
room temperature. Though large SERF magnetometer arrays have not yet been built, smaller ar-
rays should be sufficient for applications such as fMCG. In this thesis, we present the design and
characterization of a portable four-channel SERF atomic magnetometer array with a 5–10 fT/√
Hz
single channel baseline sensitivity. The magnetometer array has several design features intended
to maximize its suitability for biomagnetic measurement, specifically fMCG, such as a compact
modular design and large, flexible channel spacing from 5–15 cm. The modular design allows for
easily adding units to the array and the independent positioning and orientation of each magne-
tometer, in principle allowing for non-planar array geometries. Using this array in a magnetically
xiii
shielded room, we acquire adult magnetocadiograms and, for the first time with a SERF magne-
tometer, fMCG. We also investigate the use of different operational modes of the magnetometer to
extend its functionality, specifically modulation methods for additional directional sensitivity, the
possibility of operation in large DC fields, feedback methods to help make a high suppression ratio
hardware gradiometer, and using diffusion to perform the magnetometer measurement with atoms
outside the influence and noise contribution of pumping lasers.
1
Chapter 1
Introduction
Human biomagnetism has been an important area of fundamental research and medicine since
the first low noise measurements using superconducting quantum interference devices (SQUIDs)
[Cohen et al., 1970]. Since that time, impressive improvements have been made in the design and
engineering aspects of SQUID sensors for biomagnetic studies, and integrated commercial systems
are now available with hundreds of individual SQUID channels designed to cover an entire human
head [Wikswo, 1995]. These systems, operating with sensitivities near 1 fT/√
Hz, have allowed an
increasingly diverse set of novel measurements to be performed.
Magnetic signals can provide access to unique or complimentary information concerning un-
derlying phenomena in the human body. For example, the distribution of electrical potentials at
the surface of the skin is distorted by tissue and bone in the intervening conduction path [Wakai
et al., 2000], which makes accurate reconstruction of the original source distribution difficult and
can inhibit the electrical signals altogether. Magnetic signals from the same sources are unaf-
fected by tissue and bone, simplifying the source localization [Wikswo, 1995]. For example, mag-
netoencephalography (MEG) can be used in pre-surgical studies or to image cognitive function,
where knowledge of the position of the neurologic currents is important [Hamalainen et al., 1993,
Wikswo, 1995]. In another example, fetal magnetocardiography (fMCG) is currently the only
2
sufficiently sensitive technique for certain clinical applications, such as the detection of cardiac
arrhythmia.
Until recently, magnetometers based on superconducting quantum interference devices (SQUIDs)
provided the only option for biomagnetism applications demanding the highest possible sensitiv-
ity, usually operated in arrays of matched gradiometers with 30 or more channels. SQUIDs have
achieved a noise limit of around 1 fT/√
Hz [Robbes, 2006], but they are expensive (∼ $10 k/channel)
and require liquid 4He for the low Tc systems needed for the demanding signal-to-noise ratio (SNR)
in sensitive biomagnetic applications, like fMCG [Li et al., 2004].
Fetal MCG can be used to detect and diagnose dangerous arrhythmia in utero, such as supraven-
tricular tachycardia or torsades de pointes [Cuneo et al., 2003, Wakai et al., 2003]. In this appli-
cation, precise imaging of the fetal heart is not necessary, but electrical signals are attenuated by
an insulating layer covering the fetus, so they cannot provide sufficient resolution for diagnosis
[Wakai et al., 2000]. As a non-imaging application, fMCG demands fewer magnetometers, just
enough for signal processing enhancement through spatial filtering.
The demands of fMCG sets several restrictions on magnetic sensors. First, the the largest
feature in fMCG is the QRS peak, which has a magnitude of ∼1 pT and a bandwidth of ∼100
Hz. Therefore, the fundamental noise of the detector must be <10 fT/√
Hz in a 100 Hz bandwidth
starting near DC for a 10:1 SNR. Second, the background magnetic noise at the detector must be
below this same limit. Passive magnetically shielded rooms (MSR) using hi-permeability materials
like mu-metal are typically used, but practical limits are placed on the achievable attenuation of
external signals due to the high cost of MSR materials. Cheaper eddy current shields are suitable
at higher frequencies, but attenuated external fields in the 0–100 Hz bandwidth are often well
above the MSR noise limit in the frequency band of interest [Platzek et al., 1999]. Another way
to increase the SNR is to detect the gradient of the magnetic field, rather than the magnetic field
itself. This is the typical detection method for SQUID sensors, where detector spacing of a few
cm is chosen to maximize the SNR from the fetus compared to gradients from signals outside the
MSR or the pregnant adult’s much larger heart signal [Wolters et al., 2003].
3
1.1 Atomic magnetometer renaissance
Atomic magnetometers have recently shown promise for use in biomagnetism. Atomic va-
por magnetometers operating in the spin exchange relaxation free (SERF) regime [Allred et al.,
2002] have recently surpassed the sensitivity of SQUID systems, achieving 160 aT Hz−1/2 sen-
sitivities [Dang et al., 2010]. Additionally, the application of microfabrication techniques to the
manufacture of mm-scale alkali-metal vapor cells and integrated fiber-coupled optics [Shah et al.,
2007] allows the possibility of large arrays of SERF magnetometers, similar to current commer-
cial SQUID systems. Besides offering increased sensitivity, atomic magnetometers are operated
at temperatures between 25–180 C [Budker and Romalis, 2007], replacing expensive cryogenics
and maintenance of SQUID systems with simple resistive electrical heating and passive thermal
insulation.
Proof-of-principle biomagnetic measurements have been made previously. Adult MEG has
been measured using a similar sized fiber coupled atomic magnetometer operating as in this work,
in a different detection mode [Johnson et al., 2010]. Single channel sensitivity was comparable
to this work, but quadrant photodiode detection was used to construct gradiometer signals, rather
than the use of multiple magnetometers. A similar technique using a photodiode array was used
previously [Xia et al., 2006], but with a large non-portable setup. While using a single probe
beam to make measurements in multiple parts of the cell is appealing for technical reasons, the
effective separation between the atoms participating in the measurement is small compared the the
signal source depth of an MCG measurement. The individual sensors will measure approximately
the same signal from sources as deep as fMCG, and fMCG signals would be suppressed with the
background noise by these implementations.
Adult MCG studies have also been performed. Bison et al. [2009] used an array of 25 room
temperature Cs (non-SERF mode) magnetometers as a 19-channel set of 2nd order gradiometers
to map adult MCG in an eddy-current shield. They were able to suppress the noise by a factor of
1000 using this technique, but were still limited to 300 fT/√
Hz. Knappe et al. [2010] used a single
channel microfabricated mm-scale alkali-metal vapor cell to measure adult MCG in an MSR. Their
4
signals compare well with SQUIDs simultaneously measuring the MCG, largely because they were
able to get the detector much closer to the subject’s chest. The measurements were performed with
a sensitivity 100-200 fT/√
Hz but in a much quieter shielded room [Bork et al., 2000]. Results
from a single magnetometer were presented, as opposed to multiple channels, as we use here.
To summarize, these previous studies used either a single magnetometer, gradiometers with
baseline ∼1 cm, or magnetometers without high enough sensitivity for fMCG. In contrast, we
have developed a four-channel portable atomic magnetometer array suitable for the demands of
fMCG. The channel spacing in this work is flexible and can be adjusted between 5–15 cm. Each
magnetometer in the array features a 5–10 fT/√
Hz noise floor from 10–100 Hz and has adjustable
channel spacing and orientation. All channels are self-contained, including local tri-axial nulling
coils, in principle allowing the channels to be placed in an arbitrarily oriented non-planar geometry.
The SERF channel spacing is significantly larger than previous biomagnetic studies, suitable for
optimizing background rejection.
Using this magnetometer in a magnetically shielded room (MSR), we have demonstrated the
use of the array to successfully detect MCG from 17 adult subjects. Additionally, we have detected
the first fMCG signal using an atomic magnetometer (to the best of our knowledge), a proof-of-
principle experiment for an application well suited to the budding field of SERF biomagnetometers,
where high sensitivity but modest channel counts are required.
In addition to successfully measuring fMCG signals, we have also worked on extension of the
methods of SERF magnetometry to address shortcomings in the traditional method of operation
presented in Allred et al. [2002]. For example, SQUIDs can be made as hardware gradiometers
with excellent uniform background rejection. The most commonly used technique takes advantage
of the ability to fabricate pickup coils with very uniform characteristics for two spatially separated
turns. An equivalent technique for SERFs is important, but in many ways is a more difficult chal-
lenge. We demonstrate the use of feedback to suppress these background signals despite significant
nonuniformities in the individual magnetometer response.
5
We increase the channel count using a previously developed modulation technique [Li et al.,
2006] to simultaneously detect two magnetic field components from a single magnetometer with-
out loss of sensitivity, demonstrating the technique is suitable for use without degradation in a
closely packed array. We also make use of our relatively low buffer gas pressure cells to set up a
situation in which the atoms that participate in the magnetometer measurement do so in a spatial
region of the cell devoid of pump light, which has some practical advantages.
Finally, we describe and show the initial measurements of a method to spin-polarize alkali
atoms transverse to a strong magnetic field through a transverse spin resonance. The resonance
feature can be used to detect magnetic fields in the same way as the traditional SERF, and can be
made to be spin-exchange relaxation free despite the presence of the large DC magnetic field in
which it operates. In this transverse pumped magnetometer, the signal has the additional benefit
of being a Larmor frequency, rather than a quasi-DC signal requiring calibration (as a traditional
SERF does). This modulation method could allow SERF magnetometers to be operated in earth-
sized background fields.
The structure of the thesis is as follows: Chapter 2 will outline the difference between tradi-
tional atomic magnetometers and SERF magnetometers and outline the wide parameter space in
which the usual method of using a SERF can operate. Chapter 3 describes the details of the four
magnetometer array built and tested as the bulk of the work for this thesis. Chapter 4 will outline
the data collection procedure, including the procedures used to cancel background magnetic fields
and calibrate the magnetometers. A noise budget is presented and discussed in detail in chapter
5. In chapter 6, we discuss four extensions to the traditional DC-mode SERF magnetometer pio-
neered at Princeton by Romalis’ group, one of which may allow extension of SERF magnetometry
to high DC magnetic fields. Specific requirements of biomagnetic signals, in particular fMCG, are
dealt with in chapter 7, as well as the basic signal processing tools used to help determine magne-
tometer performance with real signals. At the end of chapter 7, we present the first SERF detected
fMCG signal. We conclude with directions for future development.
6
Chapter 2
Theory of SERF magnetometry
This chapter provides a brief overview of how an atomic gas can be used as a magnetic sens-
ing device. We provide a general description of a traditional atomic magnetometer and discuss
its limitations, followed by an introduction to the SERF regime that allowed these limitations to
be surpassed and led to the most sensitive magnetic sensing device currently developed [Dang
et al., 2010]. We will also provide an outline of the SERF magnetometer and an overview of the
theoretical description of its operation in its most sensitive and simplest mode, which we refer
to as “DC-mode.” This overview is meant to aid the understanding of the design, operation, and
characterization of the magnetometers developed as part of this thesis, and to lay the theoreti-
cal foundation for an analysis of the fine points in later chapters. We sketch or expand upon the
derivations and useful results presented elsewhere, for example in articles developing specific im-
plementations of SERFs [Allred et al., 2002, Kominis et al., 2003, Ledbetter et al., 2008], reviews
on optical atomic magnetometers [Budker and Romalis, 2007], or several thorough thesis on the
subject, in particular [Kornack, 2005, Li, 2006, Seltzer, 2008].
7
2.1 Historical optical magnetometry
Optical atomic magnetometers were first developed in the late 1950’s in the Bell-Bloom or Mx
style, which built an oscillator from an optically pumped Rb gas [Bell and Bloom, 1957, Dehmelt,
1957]. Further development led to the Hanle type magnetometer [Cohen-Tannoudji et al., 1969],
which utilized a directly pumped 3He gas intermixed with Rb gas used as a probe for the 3He
spins. Other types have been developed, such as those based on coherent population trapping
[Scully and Fleischhauer, 1992] and nonlinear magneto-optical Faraday rotation [Budker et al.,
2000]. These techniques fall under the category of optically pumped magnetometers, and have
achieved sensitivities bellow 100 fT/√
Hz [Alexandrov and Bonch-Bruevich, 1992].
We focus on the Bell-Bloom type as a simple example of an optically pumped atomic mag-
netometer. We will discuss the relaxation mechanisms that limit the sensitivity of the Bell-Bloom
magnetometer, as well as how the SERF magnetometer overcomes those restrictions.
2.1.1 Traditional atomic magnetometry
Any atom with a nonzero magnetic moment can be made to be a magnetometer. A magnetic
field applied perpendicular to the magnetic moment will cause a torque on the atom, resulting in the
Larmor precession of the atom about that magnetic field at angular frequency Ω = γB/q, where
γ is the gyromagnetic ratio and B is the magnitude of the magnetic field. For atoms, γ = gµB/~,
where g is the Lande g-factor for the level participating in the detection. The factor q is a quantity
which accounts for the hyperfine interaction and the fact that at low magnetic fields, the nuclear
spin acts to increase the inertia of the total atomic angular momentum. For a free atom, q is constant
dependent upon the nuclear spin. Since γ is determined by fundamental constants and quantum
numbers, a measurement of the Larmor frequency of a precessing atom is a direct measure of the
magnetic field.
The trick then becomes increasing the sensitivity of the measurement. As a first approximation,
the observation of precession of a collection of nV atoms with coherence time T2 can be used to
8
measure a magnetic field with precision
δB ' 1
γ√nV T2t
, (2.1)
where t is the measurement time and V is the measurement volume [Budker and Romalis, 2007,
Ledbetter et al., 2008]. When the precession is continuously observed, (2.1) is optimized for the
largest number of atoms precessing with the same phase for as long as possible. These require-
ments can contradict one another, for example when T2 ∝ n−1. Indeed, changing the relationship
between the spin coherence time and the atomic number density is the significance of the SERF
regime, and the enabling discovery for ultra-high sensitivity atomic magnetometers [Allred et al.,
2002].
Fundamentally, an atomic magnetometer requires alignment of an atomic vapor and subsequent
detection of changes in atomic alignment. For alkali vapor magnetometers, it was realized that this
could be accomplished through optical pumping and subsequent detection of the atomic precession
through the modulation of the absorbed pump light [Bell and Bloom, 1957, Bloom, 1962, Dehmelt,
1957], as in figure 2.1. Modern atomic magnetometers designed to work in a broad range of
magnetic background environments still make use of techniques developed by Bell and Bloom
(for example, [Aleksandrov et al., 1995, Groeger et al., 2006, Schwindt et al., 2007]) with technical
refinements to improve the magnetometer performance. In research conditions, sensitivities of a
few fT/√
Hz can be achieved [Aleksandrov et al., 1995], though these increases are obtained by
increasing the detector volume or operating at higher frequencies.
2.1.2 Spin exchange collisions and the SERF regime
For a fixed volume, the sensitivity of these traditional devices is often limited by spin-exchange
(SE) collisions between alkali-atoms [Allred et al., 2002], such that T2 ∼ Tse = (nvσse)−1, where
v is the thermal atomic velocity and σse is alkali-alkali SE cross section for the chosen atomic
species, (1.9 × 10−14 cm−2 for 87Rb–87Rb, [Gibbs and Hull, 1967]). When this is the case, it
is only fruitful to increase n as long as the signal-to-noise ratio (SNR) is not dominated by SE
collisional decoherence.
9
Figure 2.1: Illustration of operation of a feedback locked Mx mode magnetometer (reprinted from [Kitch-ing et al., 2011] with the author’s permission). A circularly polarized optical pumping beam creates anatomic polarization along the direction of its angular momentum. Without feedback, an oscillating fieldgenerated by the coils would drive the atoms to oscillate around B0 when the driving field was resonantwith the Larmor frequency of B0. The precession of atomic spins modifies the susceptibility of the vaporat the precession frequency, detected here by measuring the absorption of the pumping light. The systembecomes self-oscillatory at Ω0 when feedback is applied.
10
Figure 2.2: A Rb-Rb SE collision (adapted from [Walker and Happer, 1997]). These collisions preservethe angular momentum projection mf1 + mf2 , but redistribute population among the hyperfine sublevelsconsistent with angular momentum conservation. Between SE collisions, atoms in different hyperfine man-ifolds precess in opposite directions at nearly the same frequency. If they are allowed to precess for toolong, this leads to rapid decoherence as a result of the SE level redistribution.
Figure 2.2 shows a diagram of an alkali-alkali SE collision. These collisions couple the electron
spin of each colliding atom, leaving the nuclear spin of each atom, as well as the total angular
momentum, conserved during the collision time. The result is distribution of angular momentum
in a set of |F,mf〉 states to a statistical distribution of final states consistent with the preservation
of the total angular momentum projection mf1 + mf2 . Ignoring small corrections, the hyperfine
Lande g–factor for the ground state F-levels is opposite in alkali-metal atoms,
gF ' gsF (F + 1)− I(I + 1) + J(J + 1)
2F (F + 1)
= 1/2 (87Rb 52S1/2, F = 2)
= −1/2 (87Rb 52S1/2, F = 1)
So while the total angular momentum projection is conserved during a SE collisions, the redistri-
bution into different F-levels results in precession in opposite directions after such a collisions. An
atom which begins with some orientation, undergoes a SE collision, precession for a significant
portion of a Larmor period, and undergoes SE again does not end up with its orientation preserved.
In other words, SE collisions lead to decoherence if the precession angle is on the order of a single
rotation or larger (ΩTse ≥ 2π).
11
Spin exchange relaxation constitutes a fundamental limit on the coherence of an atomic gas
[Bouchiat and Brossel, 1966], and their use as magnetometers [Budker et al., 1998] for many
cases. However, it has been understood for some time that magnetic resonance lines were narrowed
when the SE rate was sufficiently high compared to the magnetic field induced Larmor frequency
(ΩTse 2π) [Happer and Tam, 1977, Happer and Tang, 1973]. Figure 2.3 shows the three relevant
SE regimes, where traditional atomic magnetometers operate in the precession regime figure 2.3(a)
and the SERF regime is shown in figure 2.3(c).
Rapid SE collisions lead to the spin-temperature distribution [Anderson et al., 1960, Baranga
et al., 1998, Savukov and Romalis, 2005], a maximum entropy distribution consistent with angu-
lar momentum and energy conservation. In spin-temperature, the density matrix can be simply
described in terms of a single spin temperature parameter β as ρ ∝ e−βFz , where
β = ln
∣∣∣∣1 + P
1− P
∣∣∣∣ (2.2)
and P = 〈S〉 /S is the electron spin polarization. This distribution relates the average total angular
momentum to the average electron angular momentum by the so-called slowing down factor q(P )
[Baranga et al., 1998, Savukov and Romalis, 2005],
〈Fz〉 = q(I, P ) 〈Sz〉 (2.3)
q(3/2, P ) =6 + 2P 2
1 + P 2(2.4)
For I = 3/2, the slowing down factor varies smoothly from 6–4 (figure 2.4a). Figure 2.4b demon-
strates how the spin-temperature distribution changes for varying polarization and the slowing
down factor for each polarization.
Rapid SE leads to a new SE relaxation regime in which the relaxation rate decreases quadrati-
cally with field [Allred et al., 2002, Happer and Tam, 1977],
ΓSE = Ω2TSE
(1
2− (2I + 1)2
2q2
). (2.5)
For 87Rb at 150 C, the SE rate is 1/TSE ≈ 70 × 103 s−1. Using this temperature and a vapor
pressure model of 87Rb [Alcock et al., 1984], figure 2.5 shows the SE relaxation rate as a function
of magnetic fields perpendicular to the pump direction.
12
Figure 2.3: Spin exchange regimes in increasing ratios of TSE/Ω, reprinted from [Happer and Tam, 1977]with author permission. Traditional atomic magnetometers operated at relatively low SE rates, as in (i).Increasing TSE/Ω initially leads to faster decoherence (ii), but then narrows again (iii) when TSE becomesthe dominant rate and enforces that individual atoms, on average, precess as the entire density matrix would.Each atom rapidly changes hyperfine levels due to rapid SE, but the net result is statistically preferentialprecession at a lower frequency, for each atom and for the density matrix as a whole.
13
(a) Slowing down factor for I = 3/2
(b) Spin-temperature distribution at various P
Figure 2.4: (a) Slowing down factor q(P ) for I = 3/2 atoms. (b) Spin temperature distributions for avariety of average polarization values. Even for zero polarization, most of the atoms (5/8) are in the F = 2manifold and rapid SE ensures the atoms maintain a weighted precession of the average spin.
14
Figure 2.5: ΓSE according to (2.5) with a range of model temperatures (120, 150, 180 C) at slowing downfactor q = 5 for 87Rb as well as comparisons at 180 C with q = 4.1 and q = 6.
15
2.2 SERF magnetometer
To be concrete, we will discuss the system of interest in this work, an isotopically pure 87Rb
atomic vapor in a buffer gas of combined He and N2. This chapter deals with the primary detection
scheme used in this work, what we call DC-mode operation, first presented by Allred et al. [2002].
The general description of a magnetometer from the previous section still applies, but the atoms
will now be interrogated by a linearly polarized probe perpendicular to the pump beam, as in figure
2.6 (this geometry and coordinate system will be used in the future equations of this work, unless
otherwise specified).
We will describe the physics of this system in the SERF limit, where rapid SE causes the
density matrix to assume a spin-temperature distribution. The ground state can be described by
Bloch equations with only a few important terms [Ledbetter et al., 2008]: optical pumping, spin
destruction, and Larmor precession. The spin destruction term will include any ground state spin
destruction mechanism, in particular coupling spin and angular momentum during binary colli-
sions, photon absorption by the probe beam, and diffusion to the cell walls.
Optical pumping creates a nonzero average electron spin along the propagation direction of the
pumping laser. Orthogonal magnetic fields cause the population spin to precess at the slowed down
Larmor frequency. Spin-relaxation collisions, the absorption of photons, and wall collisions inter-
rupt the precession. The balance between optical pumping, spin relaxation, and Larmor precession
determines the dynamics of the atomic polarization and the performance of the magnetometer.
2.2.1 Optical pumping
Optical pumping can be quite complicated in the full hyperfine picture [Happer and Wijngaar-
den, 1987]. We will use the simplifying assumption that the hyperfine levels are not well resolved
due to pressure broadening. The 6.8 GHz ground state splitting for 87Rb requires about half an
amagat or more of buffer gas in order to reach this regime at∼ 18 GHz/amg [Romalis et al., 1997].
Figure 2.7 shows the a simplified energy level diagram for 87Rb. Circularly polarized optical
pumping light resonant with the D1 transition selectively pumps atoms out of a single hyperfine
16
Figure 2.6: General magnetometer model with orthogonal pump and probe beams. The circularly polar-ized pump creates an atomic orientation, while the linearly polarized probe is sensitive to changes in thatorientation. The same coordinate system is used throughout this work.
17
Figure 2.7: Simplified energy level diagram of 87Rb. D1-line optical pumping works by utilizing angularmomentum selection rules to selectively excite atoms from a single sublevel. In the excited state the atomsare quenched by collisions with N2, which randomize the spin and populate both ground states equally.Thus, for each pump photon absorbed, 1/2 unit of angular momentum is deposited into the vapor.
level, due to angular momentum selection rules forbidding excitation from the ms = 1/2 ground
state by σ+ polarized light. Once excited, the atoms undergo spin-randomizing collisions with
the buffer gas before they undergo non-radiative decay through collisions with N2. The process
deposits 1/2 unit of angular momentum on average from the absorption of a photon. Though the
electron spin is randomized in the excited state, the nuclear spin is essentially conserved, hence the
process tends to polarize the nuclear spin as well, and creates a spin temperature distribution.
A comprehensive understanding of angular momentum optical pumping in alkali buffer gas
mixtures is still an active area of research [Happer et al., 2010, Lancor and Walker, 2010], but
for the relatively low buffer gas pressures and polarizations in these cells, the above description is
sufficiently accurate. Under these assumptions, optical pumping contributes to the evolution of the
total angular momentumd 〈F〉dt
∣∣∣∣OP
=R
2(1− s ·P), (2.6)
18
where s is the photon spin and R is the optical pumping rate (the rate at which unpolarized atoms
absorb linearly polarized light). R is the convolution of the pumping photon flux and the unpolar-
ized atomic cross-section σ0,
R =
∫ ∞−∞
Φ(ν ′ − ν)σ0(ν ′ − ν0)dν ′ (2.7)
R ≈ Φ0σ0(∆ν) (2.8)
Φ0 =Ihν
(2.9)
σ0 = refcL(∆ν) (2.10)
L(∆ν) =∆Γ/2
(∆ν)2 + (∆Γ/2)2, (2.11)
where ν and ν0 are the laser center frequencies and atomic absorption center frequencies, respec-
tively, re is the classical electron radius, f is the oscillator strength, the laser intensity is I, and
∆ν = ν − ν0 is the laser detuning from the atomic transition. The approximation (2.8) holds
when the linewidth of the atomic transition is much greater than the linewidth of the laser, and
Φ(ν ′ − ν) ≈ Φ0δ(ν′ − ν). The lasers in this work are all single mode with linewidth < 10 MHz,
and we use this approximation throughout the rest of this work.
2.2.2 Spin-relaxation from binary collisions and probe absorption
Under the assumptions for this simplified model, Rb-Rb and Rb-buffer gas binary spin-destruction
collisions are a fundamental limit on the spin-coherence time of 〈F〉 [Baranga et al., 1998, Erick-
son et al., 2000, Kadlecek et al., 2001]. The spin-destruction rates for Rb–Rb and Rb–N2 are taken
from [Chen et al., 2007], while the rates for Rb-He are taken from [Allred et al., 2002]. They are
all lumped into a single ground-state spin-destruction rate, Γsd:
ΓRb−Rb =
(4.2× 10−13 cm3
s
)[Rb] (2.12)
ΓRb−N2 =
(1.3× 10−25 cm3
s K3
)T 3[N2] (2.13)
ΓRb−4He =(9× 10−24 cm2
)v[4He] (2.14)
Γsd = ΓRb−Rb + ΓRb−N2 + ΓRb−4He, (2.15)
19
where the quantities in brackets represent the number density, in cm−3.
An additional spin-destruction mechanism is through the absorption of probe photons. Assum-
ing a weak, linearly polarized probe beam with a propagation direction perpendicular to the pump
(as in figure 2.6), we treat the absorption of probe photons as a simple relaxation mechanism. The
photon absorption rate can be written similarly to R in the limit of a narrow probe laser linewidth
compared to the atomic transition [Ledbetter et al., 2008],
Γpr = Φ0,prσ0,pr, (2.16)
where the probe photon flux Φpr has the same definition as (2.9). The atomic ensemble evolves
under the influence of spin-destruction collisions and probe absorption as
d 〈F〉dt
∣∣∣∣collisions
= −(Γsd + Γpr)S. (2.17)
2.2.3 Diffusion
Diffusion carries angular momentum from where it is deposited by the pump to other places in
the cell. Its effect on the evolution of the angular momentum is [Ledbetter et al., 2008]
d 〈F〉dt
∣∣∣∣diff.
= D∇2 〈F〉 , (2.18)
where D is the diffusion coefficient for alkali atoms under the cell conditions (buffer gas makeup,
pressure, temperature). Interactions between polarized alkali atoms and the glass container walls
tend to completely depolarize the atoms (both the electronic and nuclear spin) [Wagshul and
Chupp, 1994, Walker and Happer, 1997], enforcing the boundary condition 〈F〉|walls = 0.
There are various methods of dealing with the diffusion contribution to the evolution of the
angular momentum Ledbetter et al. [2008], Wagshul and Chupp [1994], Walker and Happer [1997].
For our purposes, the diffusion term has two important effects. 1) Broadly, it increases the rate at
which angular momentum is lost from the vapor, reducing T2 and increasing the number of pump
photons required to obtain a given steady state polarization. 2) Diffusion changes the detailed
spatial dependence of the polarization.
20
In what immediately follows, we only account for the first effect, treating diffusion purely as
an extra relaxation rate ΓD, which will be important for the estimation of the overall performance
of the magnetometer. Later, we will address the spatial dependence of all the terms involved in the
angular momentum dynamics, and include a more nuanced treatment of diffusion.
We rewrite (2.18) to describe the angular momentum loss to the walls as [Ghosh, 2009, Wagshul
and Chupp, 1994]
d 〈F〉dt
= −ΓD 〈F〉 (2.19)
ΓD ≈ D0(T )
(p0
p
)× 3
(πl
)2
, (2.20)
with the standard pressure to gas pressure ratio p0/p, and the term 3×(π/l)2 coming from a normal
mode expansion of the solutions of (2.18), and using the velocity of the atoms in the fundamental
diffusion mode. For Rb in nitrogen and helium buffer gases, the fundamental diffusion coefficients
are [Happer et al., 2010, p 193]
D0(Rb− N2) = 0.159cm2
sT0 = 60 C
D0(Rb− 4He) = 0.35cm2
sT0 = 80 C
D0(T ) = D0|T0
(T
T0
)3/2
,
where the final equation describes how to obtain the fundamental mode diffusion coefficient at a
temperature T when it has been measured at a different temperature T0. The measurements above
were both performed at p0 = 1 atm. We estimate ΓD = 420 1/s for our experiment using our cell
side length l = 1 cm, He pressure 0.029 amg, and N2 pressure 0.066 amg, at T = 145 C. This result
was obtained by finding
ΓD = q(P )
(1
ΓD,N2
+1
ΓD,He
)−1
.
In the spin-temperature limit we can express electron spin depolarization at the walls as occurring
at a rate enhanced by q(P ), since both the electron and nuclear polarization are lost at the wall.
Then the effective relaxation rate due to wall collisions is P∣∣∣diff.
= − q(P )TD
P .
21
The diffusion loss rate ΓD depends on the buffer gas makeup, pressure, and temperature, but
in a different way than the ground-state spin destruction rate due to binary collisions Γsd. Figure
2.8 shows plots of the total relaxation rate Γ = ΓD + Γsd as a function of initial helium pressure,
nitrogen pressure, and cell operating temperature. As the figures make clear, diffusion is an impor-
tant contribution to the angular momentum relaxation under our typical operating conditions, (50
Torr N2, 22 Torr He, and 145 C cell temperature). Cells optimized for maximum sensitivity would
utilize higher buffer gas pressure to limit diffusive losses.
2.2.4 Larmor precession
Between spin-destruction collisions with atoms, walls, or probe photons, the atomic ensemble
precesses due to the influence of a small external magnetic field,
d 〈F〉dt
∣∣∣∣Larmor
= Ω× 〈S〉 . (2.21)
22
(a) N2 (b) He
(c) Temperature
Figure 2.8: Total relaxation rate, including diffusion losses, for a square cell of 1 cm side length. Subfigures(a) and (b) show the total relaxation loss as a function of initial buffer gas pressures for N2 and He, respec-tively, at 145 C cell temperature. Since the relaxation rates due to diffusion and binary collisional relaxationdepend differently on pressure, a minimum exists for each buffer gas mixture. The vertical dotted line in (a)corresponds to the buffer gas pressure in most of the cells used for magnetometry in this work. Subfigure(c) shows the binary collisions depend more strongly on cell temperature.
23
2.2.5 Equation of motion and steady-state response
Under the influence of the processes listed above, and assuming optical pumping and SE col-
lisions enforce a spin-temperature distribution, then the electron polarization is governed by the
following differential equation [Allred et al., 2002, Ledbetter et al., 2008],
q(P )dP
dt= q(P )D∇2P + Ω×P +Rs− Γ′P, (2.22)
where we have assumed that the slowing down factor q(P ) is constant in time. If we fold in the
effect of diffusion using an effective total relaxation rate Γ = ΓD + Γsd + Γpr (section 2.2.3), the
steady state solution to (2.22), assuming perfect pump circular polarization s = z, is
P =R
Γ′ (Γ′2 + Ω2)
Γ′Ωy + ΩxΩz
−Γ′Ωx + ΩyΩz
Γ′2 + Ω2z
, (2.23)
where Γ′ = R + Γ. Equation (2.23) can also be expressed in terms of Pz,
Px = Pz
(Γ′Ωy + ΩxΩz
Γ′2 + Ω2z
)≈ Pz
Ωy
Γ′(2.24)
Pz =R
Γ′
(Γ′2 + Ω2
z
Γ′2 + Ω2
)≈ R
Γ′, (2.25)
where the approximations are valid when Γ′ Ω. For typical Γ′ ∼ 1000 s−1, this is true for
magnetic fields below B ∼ 1 nT. In the configuration diagrammed in figure 2.6, the probe beam
will be sensitive to Px.
To obtain a sense of the parameter space topology of a DC-mode magnetometer, it is helpful
to normalize all parameters to the total spin-destruction rate. Ignoring Γpr, which can be made
arbitrarily small, this choice of normalization is convenient because Γsd + ΓD for a single atomic
species depends on the permanent properties of the cell (e.g. buffer gas composition, cell geometry)
as well as the cell temperature, and only weakly on the cell polarization through ΓD. Per unit atom
participating in the measurement, the smaller Γ the better. Henceforth we indicate a normalized
24
Figure 2.9: Normalized sensitivity (blue) and polarization (purple-dashed) vs R. This graph indicates thatfor a given relaxation rate, the maximum sensitivity is at Pz = 1/2 and the sensitivity is maximized for anypumping rate for smaller Γ.
quantity as
R = R/Γ
Ω = Ω/Γ,
and, (2.24) is
Px =R
R + 1
(R + 1
)Ωy + ΩxΩz(
R + 1)2
+ Ω2
. (2.26)
First we examine the DC-sensitivity Px/Ωy. Assuming finite ground-state spin-relaxation, the
sensitivity is maximized for comparatively small probe absorption (Γpr Γsd + ΓD) and R = 1
(Pz = 1/2). Optical pumping is necessary to achieve any signal, but R Γ simply pins the atoms
to the pump axis through rapid absorption of pump photons, decreasing Px and the sensitivity.
Even so, figure 2.9 shows the sensitivity falls off more slowly for overly high pumping rates than
overly low pumping rates. Coupled with strong absorption from an optically thick atomic vapor,
this means beginning with R 1 and optimizing R using an applied signal is a good method for
finding the best pumping parameters.
So far, we have described the operation sensitivity of the SERF magnetometer to a single vector
projection of of Ω, and the sensitivity in figure 2.9 indicates the sensitivity to an applied field in
25
this direction. This is true for Γ′ Ωz. When Γ′ Ωz, the polarization is proportional to an
orthogonal field Px ∝ Ωx, though with diminished sensitivity. In the intermediate regime, we can
write the response to a general field perpendicular to the pump direction as (from (2.26))
∂Px∂Ω⊥
=∂Px∂Ωx
∂Ωx
∂Ω⊥+∂Px∂Ωy
∂Ωy
∂Ω⊥(2.27)
≈ R
Γ′ (Γ′2 + Ω2z)
Γ′ + ΩzΩ?√1 + Ω2
?
, (2.28)
where Ω? = Ωx/Ωy and Ω2⊥ = Ω2
x + Ω2y. The approximation in (2.28) is Ω2
⊥ Ω2z + Γ′2. Figure
2.10 shows how (2.28) (normalized to Γ) varies with Ω? and R.
A slice of figure 2.10(B), where Ωx = Ωy and approximately ideal pumping R = 1, reveals the
dependence of the maximum perpendicular sensitivity on Ωz, as well as the mixing angle of the
measured scalar response φxy,
φxy =(∂Px/∂Ωx)
(∂Px/∂Ωy)=
Ωz
Γ′. (2.29)
These quantities are shown in Figure 2.11. The importance of φxy will be seen later when we
investigate signal processing techniques and gradiometer performance using multiple independent
magnetometers (chapter 4 and chapter 6).
Another interesting feature of the DC-mode magnetometer is its response to an adiabatically
(compared to Γ) swept Ωy field. When the other fields are well nulled, (2.24) has a dispersive-like
shape,
Px =RΩy
Γ′2 + Ω2y
. (2.30)
The shape of this feature varies interestingly as non-zero bias fields Ωx,z are introduced, and some
examples are shown in figure 2.12a.
Ideally, the curves in figure 2.12 could be used to characterize the pumping rate and transverse
field magnitudes of the magnetometer. In reality, this can be complicated. The spatial dependence
on the pump laser becomes important, as R→ R(x, y, z). The pump exits a single mode fiber be-
fore going through the cell (see chapter 3) and so has a Gaussian profile. If the waist is known, the
transverse dependence on R(x, y) is straightforward, at least at the front of the cell. The longitudi-
nal dependence R(z) is more complicated and depends on Pz. Even in the case of zero transverse
26
Figure 2.10: DC sensitivity to fields perpendicular to the pump, Ω⊥. The most sensitive operation point iswith Ωz = 0 and Ω⊥ = Ωy (A). However, residual light shifts can impart nonzero Ωz , which increases therelative sensitivity to Ωx, while decreasing the overall maximum sensitivity, as in (B) and (C).
27
(a) (b)
Figure 2.11: (a) Transverse sensitivity at R = 1 and Ωx = Ωy (black) and the individual componentsensitivity to Ωx (red dash-dot) and Ωy (blue, dashed)). (b) Mixing angle, describing how much responseis from each transverse magnetic field component. φxy = 0 corresponds to response from Ωy alone. Themixing angle is somewhat suppressed at higher pumping rates.
28
(a) Swept Ωy at various R and other fields zero (b) Swept Ωy , R = 1, single nonzero transverse field
(c) Swept Ωy , R = 1, Ωx = 1 (d) Swept Ωx, R = 1, Ωy = 0
(e) Swept Ωx, R = 1, Ωy = 1 (f) Swept Ωy , with R = 1 and transverse fields zeroed
Figure 2.12: Plots of the atomic response to an adiabatically swept magnetic field under various conditions.Plots (a)–(c) show swept Ωy, while (d) and (e) show the response to swept Ωx. When the transverse fieldsare swept, the Pz is changed significantly (f). Practically, this alters the propagation of the pump beamthrough the cell, making characterization by swept field a more complicated problem.
29
fields and no light shifts, (2.30) is now spatially dependent and the resulting experimental mea-
surement is an average over the pump and probe intersection volume. We do not account for these
effects in presenting these curves.
For these reasons, experimentally obtained data corresponding to figure 2.12 is easiest to in-
terpret when a small section of the central feature is used, where the applied field is small and
∂Pz/∂Ωy ≈ 0. In this central regime (which is, after all, the DC-SERF operating regime), the
slope of the experimental signal is a figure of merit, provided probe beam parameters are constant.
That is, a magnetometer with a sharper slope will have a higher fundamental sensitivity (per unit
atom). We will deal more with the spatial dependence of the pump laser in section 2.4.
2.2.6 Response to dynamic fields
We now consider the response of (2.22) to dynamic magnetic fields. Imagine both DC and AC
fields are present, Ω = Ω0 + Ω1eiωt, where Ω1 Ω0. Then the AC fields do not much change the
steady-state solution, and P0 is still (2.23). If P(t) = P0 + P1(t)), then (2.22) gives
q(P )P1(t) = Ω1 ×P0eiωt + Ω0 ×P1(t) + Ω1 ×P1(t)eiωt − Γ′P1(t) (2.31)
+ Ω0 ×P0 +Rs− Γ′P0︸ ︷︷ ︸0
, (2.32)
where the last terms sum to zero because P0 is the steady-state solution. The first order linear
response at the applied frequency is then
−iωP1 = Ω1 ×P0 + Ω0 ×P1 − Γ′P1, (2.33)
where ω = q(P )ω. Solutions to (2.33) can be found by transforming into the rotating basis,
e.g. P± = Px ± iPy, but are complicated. An important simplification is when Ω0 = Ω0z z and
Ω1 = Ω1xx+ Ω1yy. Then
P1x = P0z(Γ′ − iω) Ω1y + Ω1xΩz0
(Γ′ − iω)2 + Ω20z
(2.34)
|P1x| = P0z
√(Γ′2 + ω2) Ω2
1y + Ω20zΩ
21x
(Γ′2 + Ω20z)
2+ ω2 (ω2 − (Ω2
0z − Γ′2)). (2.35)
30
A final useful simplification is when Ω0 = 0, in which case (2.35) gives the frequency response
for the zero field magnetometer
|P1x| = P0zΩ1y√
Γ′2 + ω2. (2.36)
In the zero bias field regime, the SERF magnetometer acts as a first order low pass filter centered
at zero frequency, and is sensitive only to the y-component of the oscillating field. The bandwidth
would seem to be Γ′/q(P ), but the amplitude of P0z depends upon similar factors, and is a subtlety
that will be addressed shortly (section 2.2.7).
Figure 2.13 shows the amplitude and phase response for various scenarios of operating param-
eters. The two cases described explicitly in (2.35) and (2.36) are the most important for normal
DC-mode operation.
31
(a) Response to Ω1y when Ω1x = 0, Ωz = 0− 8 (b) Phase response for (a)
(c) Response to Ω1x when Ω1y = 0, Ωz = 0− 8 (d) Phase response for (c)
(e) Response to Ω1x = Ω1y = 1/√
2, Ωz = 0− 8 (f) Phase response for (e)
Figure 2.13: Amplitude (a,c,e) and phase (b,d,f) response of the magnetometer for differing projections ofan AC field onto the x-y plane, and for different DC fields applied along the pump direction. The variouscurves all correspond to the same set of Ωz , as in (a).
32
(a) Relative response of Ω1x to Ω1y at Ωz = 1− 8 (b) Phase difference in response
Figure 2.14: (a) The relative amplitude of the magnetometer sensitivity to Ω1x and Ω1y depends upon theapplied Ωz , with increasing relative (though not absolute) sensitivity to Ω1x increasing from zero. (b) Thephase difference between the response to orthogonal magnetic fields.
33
2.2.7 Sensitivity and Bandwidth
The normalization used in the previous sections is useful for visualizing the behavior of a SERF
magnetometer in DC-mode as experimental quantities are varied. However, normalization by Γ
obscures some important considerations for applied use, such as how the magnetometer frequency
response overlaps with the signals we wish to detect, e.g. MCG. It is clear from (2.35) that the
effective bandwidth depends upon the three parameters R, Ωz, and Γ. Yet how these parameters
should be varied in order to obtain the greatest SNR for a MCG is unclear.
Using (2.35), we can obtain
∂ |P1x|∂Ω1y
∣∣∣∣Ω1x=0
=P0z
Γ
√(R + 1)2 + ω2
((R + 1)2 + Ω20z)
2 + ω2(ω2 − Ω20z + (R + 1)2)
, (2.37)
which unambiguously decreases with Γ, though we should note again this is a per atom quantity
and does not take into account changes in total sensitivity due to atomic number density.
The dependence of the magnetometer response to R and Ω0z is more complicated. To provide
a concrete description of the frequency response of a magnetometer, we model a 87Rb vapor cell at
145 C with 50 Torr N2 and 22 Torr He (the estimated parameters for our cells, see chapter 3). Evalu-
ation of (2.12) and (2.20) for these conditions gives Γsd = 2π×8.1 Hz and ΓD = 2πq(P )×13.1 Hz.
Using the sum of these two terms as the base relaxation rate Γ0, we model the magnetometer un-
der Γ = Γ0 and Γ = Γ0/2. Figure 2.15 shows the relative response under several conditions,
normalized to (2.35) evaluated at Γ = Γ0, R = Γ0, and Ωz = 0,
P1x|ref. =Ω1y√
2√
4 + q(1/2)2ω2.
Relative response above unity implies better sensitivity than P1x|ref. (again, per unit atom). In
chapter 7 we will investigate the optimum operating conditions for the magnetometer given what
is known about the frequency content of an MCG and fMCG signal. A brief inspection of figure
2.15 seems to indicate that the high frequency response can be obtained at the smallest expense of
low frequency response by increasing the pumping rate.
34
(a) Response to Ω1y , normalized to the response when Ωz = 0, R = Γ0
(b) Response to Ω1y , normalized to the response when Ωz = 0, R = Γ0
Figure 2.15: Both graphs show the magnetometer response, normalized to the case when R = Γ0 and thereare no applied DC fields. Subfigure (a) emphasizes that decreasing Γ globally increases the sensitivity atevery frequency for the same relative pumping rate. Subfigure (b) demonstrates the interesting trade-offbetween sensitivity at different frequencies. Black (red) lines represent R = 1(2), and solid (dashed) dot-dashed lines represent Ωz = 0(1)2. The high-frequency relative response is determined by the pumpingrate. The relative low-frequency limit, as well as the details of the transition between low-frequency andhigh frequency relative response depend upon Ωz . In these plots, the dependence of q(P) on R is accountedfor.
35
2.2.8 Probe Faraday rotation
The previous sections have described how the pump laser and the magnetic fields in figure 2.6
interact with a Rb vapor cell to produce a sample polarization sensitive to magnetic fields. The
detection of these magnetic fields can be accomplished using any probe sensitive to changes in P.
A simple and very sensitive method utilizes the fact that P alters the atomic polarizability, which
can be measured through the detection of a suitable parameter of a laser propagating through the
vapor. This interaction is characterized by the dipole electric polarizability↔α. Expressions for the
dipole electric polarizability are derived following [Bonin and Kresin, 1997] in appendix A. We
use the results here.
For the D2 transition using the assumptions made in appendix A, the Faraday rotation is
φ = −1
4[Rb]lrefD2cD(∆ν)Px. (2.38)
Absorption of the probe beam has two effects. Higher absorption rates decrease sensitivity through
Γpr (2.16). Absorption also decreases the number of photons used to measure the rotation angle,
decreasing the signal size and increasing the effective photon shot noise limit of the magnetometer.
While (2.38) is maximized at D(∆ν)|max → D(∆Γ/2) = 1/2, the optimum probe detuning is
actually ∆ν > ∆Γ/2 due to these two effects. Ignoring for a moment Γpr, the signal size S ∝ I0φ.
Assuming exponential absorption and Px 1, this can be written
S =P2
2xOD(x)e−OD(x)By
B0
(2.39)
where x = ∆ν/(∆Γ/2) is the normalized detuning, OD(x) = nlσ02 is the optical depth, and
B0 =
(Rγ
Γ′2 + Ω2
)−1
(2.40)
is a characteristic effective field, which is determined by Px = By/B0 for small Ωx. Figure 2.16
shows how absorption modifies the maximum signal size.
In DC-mode, φ is directly proportional to the applied magnetic field through the frequency
dependent Px (2.35), and a direct measurement proportional to this angle can easily be achieved
for small angles using a balanced polarimeter. For reference, using our experimental parameters
36
Figure 2.16: Signal size versus detuning from the D2 transition, normalized to the transition linewidth. Thedashed line is the purely dispersive shape in the rotation angle, while the solid red line takes into accountabsorption from the D2 line as well. The model uses the standard cell parameters from appendix B.
37
(see appendix B), (2.38) evaluates to φ/By ≈ 0.1µRad/fT. For perspective, the differential pho-
tocurrent is ∆I = 2P0εφ, with probe laser power P0 = 1 mW and a photodiode power to current
conversion ε ≈ 0.6 A/W, is around 250 pA/fT.
38
2.3 Warts and imperfections
The discussion of section 2.2 is meant to provide a feel for the primary operating regime of
the magnetometer (DC-mode) as well a basis for understanding the experimental setup and trade-
offs made to build an array of magnetometers. The SERF magnetometer model discussed above
ignores many important aspects of magnetometer performance. Some of these were mentioned
explicitly, such as ignoring the effects of diffusion, the spatial dependence of the laser profiles,
and the light shift from the pump lasers (which causes an effective magnetic field). We discuss
these additional complications in the final part of this chapter. The light shift especially will have
important bearing on the use of the magnetometers in an array. We will generally ignore the
hyperfine splitting in this work. It mainly affects tunable parameters like the pumping rate for
a certain absolute laser frequency, and ignoring it makes most of our calculations approximate,
especially in the frequency region between the hyperfine levels. The simulations and experimental
work are mostly performed with both lasers many linewidths off resonant where the fine-structure
description should be sufficient.
2.4 Spatial dependence of laser parameters
The spatial dependence of the pump and probe lasers is important in understanding and opti-
mizing the magnetometer design. The transverse intensity profiles of both lasers are well described
by Gaussian profiles at the front of the cell. As they propagate through the atomic vapor, absorption
complicates the dependence on the longitudinal coordinate.
2.4.1 Transverse spatial dependence of a Gaussian beam
Since both lasers exit a polarization-maintaining fiber before interacting with the atoms (see
chapter 3), both lasers are Gaussian. The intensity of a general Gaussian beam propagating in the
z direction with minimum waist w0 can be written as a function of position,
I(r, z) = I0
(w0
w(z)
)2
exp
(−2r2
w2(z)
), (2.41)
39
where the intensity at the center of the beam is
I0 =2P0
πw20
, (2.42)
P0 is the total laser power and w0 is the beam waist. The parameter w(z) describes how the
transverse profile of the beam changes as a function of propagation distance. It can be expressed
in terms of the Raleigh range zR
w(z) = w0
√1 +
(z
zR
)2
(2.43)
zR =πw2
0
λ. (2.44)
Various w0 have been used during this work, but all all are larger than 500 µm. In other words, for
all lens configurations used, the collimated beam zR > 1 m. The optical path length here is on the
order of 10 cm, so w(z) ≈ w0. Then we can write the intensity at the front of the cell for each laser
(using the coordinate system in figure 2.6)
I1(x, y, z = 0) = I10 exp
(−2
x2 + y2
w21
)D1 Pump (2.45)
I2(x = 0, y, z) = I20 exp
(−2
y2 + z2
w22
)D2 probe, (2.46)
where we have assumed the center of the lasers are centered on the cell, and the center of the cell
has coordinates (x, y, z) = 1/2× (l, l, l), where the cell is a cube of side length l. Note the naming
convention, where Ai parameters denote pump (probe) values for i = 1(2), respectively.
2.4.2 Longitudinal dependence of a weak probe
Beer’s law describes the attenuation of light as it passes through an absorbing medium. For
intensity I and propagation direction x, the absorption is
dI
dx= −nσI. (2.47)
For a relatively weak probe beam (Γpr R), the polarization is independent of the probe beam.
Assuming Px 1, σ ≈ σ02(∆ν), and the solution to (2.47) can be written by inspection,
I2(x) = I02e−nσ02x Probe absorption. (2.48)
40
The total probe intensity profile is the combination of (2.48) and (2.46),
I2(x, y, z) = I02e−nσ02x exp
(−2
y2 + z2
w22
)Probe intensity profile. (2.49)
2.4.3 Longitudinal dependence of the pump
By design, the pump beam optically pumps the atoms, which makes the absorption power
dependent (nonlinear). Further, collisions with the wall depolarize the atoms, imposing a boundary
condition for the polarization equation. Finally, as discussed in section 2.2.3, diffusion alters the
spatial dependence within the cell of the pump beam.
The differential absorption equation is the same as for the probe case, only for the pump the
cross section depends upon the polarization. For the pump beam propagating in the z direction
with circular polarization, the differential absorption is
dI1
dz= −nσ0D1(1− Pz)I1. (2.50)
When the laser linewidth is small compared to the absorption linewidth, the frequency profile
of the laser is not shifted as it is absorbed (no spectral hole-burning) [Lancor et al., 2010], and
R(z) ∝ I1(z), sodR
dz= −nσ0D1(1− Pz)R. (2.51)
Taking full measure of these effects requires the simultaneous solution of (2.51) and the steady-
state of (2.22), with the boundary conditions P (r)|r→walls = 0 and initial condition R(x, y, 0) =
R0(x, y), where R0 is described by the intensity profile of the Gaussian beam, as discussed above.
A full treatment requires numerical solution to the equations. However, much of the physics can
be captured using various simplifications and approximations.
To start with, we will use a simple model to account for diffusion in the single dimension
longitudinal to the pump [Walker and Happer, 1997]. The boundary condition for the pumping
rate is R(z → 0) = R0, and we enforce the diffusion depolarization boundary condition by the
41
approximate solution
P (z) ≈ P0(z)(1− e−z/zD
) (1− e−(l−z)/zD
)(2.52)
P0(z) =R(z)
R(z) + Γ. (2.53)
Care must be taken in treating the diffusion relaxation rate. If we want to get reliable estimates for
particular cell configurations, our model must account for all the angular momentum loss due to
diffusion. The angular momentum loss in the longitudinal direction should fall out of the solution
to the problem, but the loss in the transverse plane must be explicitly included. We do this phe-
nomenologically by setting the Γ = Γsd + ΓD|2D, where the 2D loss rate is simply ΓD|2D = 23ΓD.
The parameter zD is a characterization of the longitudinal distance travelled by the angular
momentum through diffusion in time TD ∝ 1/(R + Γ),
zD = 2π
√q(P )
D
R + Γ, (2.54)
where D is the diffusion coefficient discussed in section 2.2.3, and Γ includes transverse diffusion,
as described above. Then (2.51) can be dealt with iteratively in small steps, beginning at the
front of the cell, ensuring modest changes in pumping rate at each step. Figure 2.17 shows the
diffusion length as a function of pumping rate (R = R/(Γsd + ΓD)) and the expected longitudinal
pump profile and polarization profile for various pumping rates, with the cell parameters otherwise
similar to those described above. We use an approximate effective pumping rate,
Reff =Rfront +Rback
2
Reff =
R0 − 12
R0 ≥ 12
0 R0 <12
to calculate an average slowing down factor and transverse diffusion relaxation rate.
Figure figure 2.18 shows solutions for many different combinations of initial normalized pump-
ing rate R and pump detuning. These plots are modelled at the same cell parameters discussed
above. The pump power was set to 4 mW, a value approximately consistent with the maximum
value delivered to four cells simultaneously. A laser waist was chosen (to vary the intensity) as
42
Figure 2.17: The diffusion length zD from the simple model of Reff and the operating parameters usedthroughout this chapter. The length is characteristic of how far angular momentum is carried from its originin a time characterized by the rate R+ Γsd + 2/3× ΓD, where the diffusion term is reduced to account forrelaxation in the transverse direction. The labelled point is at an initial pumping rate equal to the collisionalspin-destruction rate.
well as an initial pumping rate R. Then the detuning was chosen to make these values consistent.
Clusters of curves of the same color in figure 2.18 correspond to the same initial pumping rate but
different intensity and detuning. The table provides a list of the particular parameters used, where
the detuning is expressed in terms of number of linewidths. For some of the lower intensities, high
pumping rates are nonphysical for the pump power. We will expand on this simple 1-D model in
chapter 6, where we will discuss operation of the magnetometer in a regime where the pump waist
is much smaller than a diffusion length.
43
Figure 2.18: Solutions to the coupled pump 1-D pump propagation and polarization equations for variouspumping parameters. Clusters of the same colored curves in the pumping plot correspond to the same initialR but at different combinations of intensity and detuning. The resulting polarization as a function of longi-tudinal coordinate is shown below. The laser power was set to 4 mW, and a waist and initial pumping ratechosen. Then the detuning consistent with those parameters was calculated. The table lists the parametersused in the calculation.
44
2.5 Light shift
The atomic polarizability has a vector component that gives rise to an effective magnetic field
[Appelt et al., 1999, Bonin and Kresin, 1997, Happer and Mathur, 1967]. In appendix A we present
a derivation that follows [Bonin and Kresin, 1997] for finding the vector light shift. The result from
a circularly polarized pump beam near the D1 resonance, as it pertains to atomic magnetometry,
can be found in the literature [Shah and Romalis, 2009b],
ΩLS = refcΦ0D(ν)z (2.55)
= R∆ν
∆Γ/2z, (2.56)
where (2.56) is obtained for the narrow laser linewidth limit, using (2.8). This field acts as any real
Ωz in how it affects the response of the magnetometer, though ΩLS → ΩLS(x, y, z) through the
complicated Gaussian and propagation determined spatial dependence of the pump beam as it is
attenuated by the atoms.
Figure 2.19 shows a plot of the light shift in field units ΩLS/γ, normalized to w0 = 1 cm and
P0 = 1 mW. Since ΩLS ∝ P0/w20, figure 2.19 can be used to quickly estimate the ΩLS for any input
pump beam parameters. With γ ≈ 2π×28 GHz/T, the resulting effective field is larger than the spin
relaxation mechanisms in the sensitivity equation (2.35). This would cause a noticeable decrease
in sensitivity and change in frequency response if the background magnetic fields were cancelled
but the light shift remained. In reality, the procedure used to cancel the background magnetic
fields cannot distinguish between light shifts and real magnetic fields, and the signal weighted
average light shift is cancelled. Gradients in the light shift will remain, mainly decreasing the
magnetometer sensitivity by decreasing the sensitivity at the spatial locations in the cell where the
pump intensity is high enough to cause the local light shift to be much different than the average
light shift.
Finally, the light shift has important bearing on using more than one magnetometer. If the
magnetometers and pump polarization are not aligned so the light shifts are pointed in the same
direction in the room reference frame, there can be effective magnetic field differences of the order
of the light shifts at the two magnetometer positions. This fact makes the use of a single set of
45
Figure 2.19: The light shift from the cell with buffer gas contents described throughout this work, normal-ized for unit power (mW) and laser waist (cm). Comparing this figure to figure 2.18, it is clear that for manysituations, the light shift can cause sizable magnetic fields. For example, with a reasonable w0 = 0.25 cm,the peak light shift is 100 nT for a 1 mW beam. A 0.25 cm waist essentially ensures that the gradient of thelight shift will be the entire light shift, since the transverse intensity profile varies widely over the cell area.
46
cancellation coils to cancel the ambient fields difficult, since one set will not be able to cancel the
light shift at both magnetometers.
47
Chapter 3
Apparatus
One of the main goals of this work was the design and implementation of a device for mea-
suring actual magnetic fields from human hearts which could be set up quickly and temporarily in
a magnetically shielded room (MSR). This loosely puts two competing sets of design constraints
on the magnetometers. For the optimum measurement capabilities, those constraints come down
to maximizing the SNR from a heart. As for the magnetometer as a practical device to be used
with human subjects, the limitations have to do with flexibility in positioning the magnetometer,
human comfort, and other practical matters. Table 3.1 summarizes these requirements, as well as
the way they are addressed in our design. The rest of this chapter will expand upon the details of
the design, the challenges faced and the ways in which these challenges were overcome.
48
Table 3.1: Summary of the design requirements for an array of multiple SERF magnetometers for mea-surement of biomagnetic signals. Requirement type refers to either requirements relating to maximizing theSNR of a biomagnetic measurement or the practical necessities of operating with human subjects, with parttime access to an MSR, and with the understanding that the magnetometers are still being prototyped andmust retain some flexibility in configuration.
Req. type Requirement Design solution
SNR
Minimize working cell to skin distance Good thermal insulation (6 mm skin to cell).
Minimize metal near cellUse non-magnetic ceramics and plastics forstructural parts, ensure metals are copper oraluminum, if possible.
For array, make unit spacing ∼5 cm for fMCG(baseline similar to source distance)
Multiple cells, enforces single magnetometersize limitation.
Practical
Array must be portable
Completely modular design:• Pump and probe single lasers fiber cou-
pled to all magnetometers.• Individual cubic tri-axial coils for each
magnetometer for local field adjust-ment.
• Electrical heating.Optics and electronics fit on a wheeled cart, canbe installed in minutes.
Array must be re-positionable with respect tostationary subject laying on a bed
Complete modularity, and mount to commer-cial SQUID, which has ceiling gantry.
Flexibility to make on-the-fly changes Magnetometer optics are reconfigurable.
49
3.1 Cell, heatsink, and heaters
Figure 3.1: Single magnetometer unit (left) with view inside the thermal insulation (right). Cubic tri-axialnulling coils are wrapped around the insulation. The optics are removable and completely reconfigurable.We have found the 4 angle greatly reduces probe noise, we presume by reducing etalon effects.
Figure 3.1 shows a single magnetometer unit. The inset displays a Pyrex 87Rb vapor cell press-
fit inside a boron-nitride heatsink thermally contacted to resistive-film heaters and a high temper-
ature thermistor, all surrounded by thermal insulation. The vapor cell is 1 x 1 x 5 cm3 rectangular
container with isotopically pure 87Rb. For this work, four different cells were used, three from
Triad Technologies (nominally channels 1, 2, and 4) and the other from Opthos Instruments (chan-
nel 3; we use “channels” and “magnetometers” interchangeably throughout this work). The Opthos
cell was filled with 100 Torr N2 buffer gas, while other cells contained 50 Torr N2and 760 Torr He
at the time of manufacture. The N2 prevents radiation trapping [Happer, 1972], while the He was
included to limit diffusion to the walls.
Subsequent absorption linewidth measurements of the Triad Technologies cells showed slow
loss of buffer gas pressure indicating He loss consistent with diffusion through the Pyrex cell
walls [Walters, 1970]. While unanticipated, this is a well known problem of borosilicate glasses
(such as Pyrex), and its prevention requires the use of aluminosilicate glass (GE-180, for example)
commonly found in high pressure SE optical pumping experiments. The pressure in the cells can
50
be estimated as a function of time by [Walters, 1970]
p = p0e− AV lPt, (3.1)
where p0 is the original cell pressure and A and V are the cell area and volume, l = 1.25 mm is
the cell wall thickness, and P = 1.75× 10−9 cm2/s is the diffusion coefficient for Helium through
Pyrex. This results in a time constant of around 170 days for our cell parameters. For reference,
the three cells containing Helium were received in mid-March 2010. Finally, it bears mentioning
that although cells 1,2, and 4 are nominally the same, they seem to exhibit different characteristics
from one another, and obviously from cell 3. More information about the cells can be found in
appendix C.
The cell is mounted in a thermally conductive ceramic heatsink. The clamping mechanism
holds the heat-sink approximately square to the pump and probe laser beam paths. If the heat-sink
also held the cell square to its sides, this would maximize etalon effects from the surfaces of the
Pyrex cell. While in theory this design should be insensitive to these kinds of effects, we have
found a substantial difference in the differential noise of a transmitted probe through an empty cell
when the probe is sent through perpendicular to the cell faces versus at a small angle. Thus, the
heat-sink is machined to hold the cell at a 4 angle to the laser paths.
Aside from holding the cell, the main purpose of the heatsink is to thermally contact the cell
with the electric heaters, yet maintain separation between them to limit the size of the heater mag-
netic fields produced at the cell detection volume. The ceramic should have the highest thermal
conductivity possible in order to minimize the power needed by the heaters and the insulation
thickness. We have used two machinable ceramics with success: Shapal-M (thermal conductiv-
ity of 90 W/m K) and Boron-Nitride (30-60 W/m K, see appendix D). The heatsinks for all four
magnetometers used here are Boron-Nitride, though Shapal-M is preferred for its higher thermal
conductivity and because Boron-Nitride is self-lubricating and tends to leave a thin film on contact
surfaces. Additionally, Boron-Nitride heat conduction is anisotropic, so a rod may conduct heat
radially more than axially, depending on the orientation of the ceramic molecular axis.
The maximum use temperature for a SERF Rb magnetometer is∼180 C, requiring special care
in material selection. Many materials were considered for the insulation, but the current insulation
51
is Aerogel, a silica based carbon-fiber reinforced fabric, with an ultra-low thermal conductivity
of 0.021 W/m K (for reference, the thermal conductivity of still air is 0.024 W/m K). Using three
layers of 2 mm thick sheets to surround the heat sink, with holes cut for optical access, allows
maintaining the cells at 140–180 C with about 4 W of heater power and warm but not uncomfort-
able outer surface temperature. Assuming this system obeys Newton’s law of cooling, we can
estimate the effective thermal conductivity of the Aerogel insulation system including loss from
the openings necessary for the laser beam path. In steady-state, the heat loss from the ceramic to
the environment should go as
heffA(∆T ) = Ph, (3.2)
where heff is the effective heat transfer coefficient from the ceramic heatsink to its surroundings,
A is the ceramic surface area, ∆T is the temperature difference between the ceramic and the
surroundings, and Ph is the heater power required to maintain ∆T . Evaluating using the ceramic
surface area as if it were solid, heff = 4.6 W/(m2 K). We can use a lumped capacitance model
of the heat transfer to write an effective thermal resistance Reff and compare it with the thermal
resistance we would expect from heat diffusion through the thermal insulation Rt,
Reff =1
heffA
Rt =LtktA
where Lt and kt are the thermal insulation thickness and conductivity. The enlightening ratio
Rt/Reff = 1.3 indicates the actual power loss is enhanced by about a factor of 1/3 from the win-
dows over what it would be where the entire magnetometer wrapped in thermal insulation.
The time-constant for the system of density ρ, specific heat Cp and volume V to come into
thermal equilibrium is
τ =ρCpV
heffA=ρCpV∆T
Ph
, (3.3)
where the second equality uses (3.2). This is around 40 minutes for our Boron-Nitride heat sink.
52
(a) Mirror mount (b) PTFE window tubes
Figure 3.2: (a) Mirror mount for steering the probe beam. A single screw can be set in either the throughhole (with a threaded hole in the block beneath) to pull the flexure hinge and tilt the mirror downwards, or inthe threaded hole to push against the block below and tilt the flexure hinge and mirror upwards. The mirrorsits on a mount attached to the flexure hinge with a threaded rod so it can be rotated about its axis. (b) Adiagram of four window tubes. There are staggered insets for two different diameter windows (10 mm and1/2 in). The windows have a tendency to fall out, but can be kept in using FEP tape stuck along their outerrim to provide pressure between the tubes and the window edges.
53
3.2 Optics and mounts
Cuts must be made in the thermal insulation to allow optical access. Heat loss through these
cuts is limited by Teflon tubes that are loosely fitted against the heat sink in the insulation, and
which hold two AR-coated windows separated by a thin air gap, as shown in figure 3.2(a). In
addition to providing better thermal insulation, these tubes provide a clear optical path maintained
free of debris from the stringy/powdery Aerogel cloth. We have noticed occasional (building up
over months) deposits on the windows of a white, dusty substance. The source of this material
is probably white PTFE thread-tape that we have used to clamp the heatsink pieces together, and
which we have found to bleach over time. Further investigation revealed PTFE tapes have varying
maximum use temperatures and not all are suitable for the SERF oven. The optics tubes were
designed to be removable for cleaning as a result of this buildup. However, we have some initial
indication that the magnetometers perform just as well without the optics tubes, and it would be
interesting to see if they are truly necessary in the future.
Figure 3.2(b) shows the plastic mirror mounts which enable steering the probe beam through
the cell and detection optics. Rotation is accomplished by turning the mounts on a threaded rod,
while standard screws tilt the mirror by pulling/pushing the plate holding the mirrors to/from the
mirror mount base. The base and mirror plate are attached through a flexure hinge. Mounting is
eased by the use of rectangular prism shaped dielectric mirrors.
Figure 3.3(a) shows a schematic of the optical elements used to operate the magnetometer.
The pump beam is collimated by a single lens or lens system to achieve the desired pump diameter
(collimated beam waists from 0.5–2.4 mm were used in this work), then sent through a Glan-Taylor
polarizer aligned with the output of the polarization maintaining fiber to maximize the transmitted
light. Finally, a 0-order quarter waveplate is used to circularly polarize the output beam. Figure
3.3(b) shows a CAD image of all the optics in the schematics of figure 3.3(a). Both lasers are
fiber-coupled from sources outside the MSR. The pump beam path goes through the central 1”
bore (tapped with Thorlabs’ 1.035”-40 threads).
54
(a) Optics schematic
(b) CAD drawing of optics
Figure 3.3: (a) Schematic of the optics necessary for operation of the magnetometer, including creatinga circularly polarized pump and a linearly polarized probe with balanced polarimitry on the probe. (b)CAD image of the optics tubes, which is transparent to show the optics mounted inside. Tubes are meantto be compatible with Thorlabs parts, using Thorlabs threading (either 0.535-40” or 1.035-40”). All partsare either mounted in externally threaded adapters, or are held in place by retaining rings. The opticsconfiguration shown here produces w0,pu = 2.5 mm, w0,pr = 1.1 mm
55
There are two 0.5” collimation tubes (tapped with Thorlabs’ 0.535”-40 threads). The output of
the probe polarization-maintaining fiber is collimated by a single lens or lens system (collimated
beam waists of 1.2–2.35 mm were used in this work) and sent through a linear polarizer. After
being reflected through the cell by the two mirrors in figure 3.2(b), there is a 0-order half-waveplate,
a Wollaston polarizing beam splitter, a focusing lens, and the photodiodes wired to produce a
difference signal.
The fibers are single-mode polarization maintaining (PM, terminated in FC-APC connections).
We have found these connections can become magnetized, and periodic demagnetization has sig-
nificantly decreased the residual magnetic fields at the cell sensor volume. Additionally, the polar-
ization of the lasers gets rotated in the fibers by 1/100-1/1000. This would look like a signal if not
for the polarizer in the probe path on the output of the probe fiber. We fully discuss this effect in
chapter 5.
3.3 Electronics
A single atomic magnetometer operating in DC-mode has four separate electronic subsystems:
cell heaters and thermistor, nulling/calibration coils and coil driving circuitry, and the photodiodes
with transimpedance amplifiers, and an analog to digital converter (ADC) acquisition system.
The electronics requirements for an atomic magnetometer operated in a MSR present a unique
set of design challenges. At typical calibrations, 1 fT/√
Hz level sensitivity requires detection of
pA/√
Hz level electrical currents, sometimes near DC. This task is complicated by the requirement
that detection electronics not generate excess large gradient DC fields or excess magnetic field
noise. Additionally, the magnetometer must be portable (to get it out of the way when the MSR
needs to be used for SQUID measurements) and actual biomagnetism measurements require it to
be flexibly positioned.
The easiest way to satisfy most of the requirements is to use long flexible electrical cables and
to put the electronics outside the MSR. This is the tact we have taken in this work, yet it is at odds
with generally accepted low noise analog electronics techniques, which preference shorter cable
56
Figure 3.4: (a) Illustration of the current path in the cell heaters. (b) Putting two heaters back to back andwiring them in anti-series helps suppress the magnetic field produced by heater currents. Each cell uses twosets of these heater pairings, wired in series to reduce the magnetic field further.
lengths, especially for photodiode leads. The electronics components described here indicate a
balance of these competing objectives.
3.3.1 Heaters and thermistors
We begin with the heaters and thermistor. The cells were operated at a temperature of 140–
180 C. Each magnetometer requires 4 W of heater power in thermal equilibrium at 145 C, corre-
sponding to electrical currents near 0.5 A, large enough to produce potentially disruptive magnetic
fields and gradients.
The heaters are Minco thermofoil (model #HR5578R4.6L12A), covered in a silicone coating.
Each heater has a resistance of 4.6 Ω and is a 19.05 x19.05 mm2 square, with a small protrusion for
the heater leads. The heater foil is patterned in a 2D plane (see figure 3.4). This allows matching
two pairs of heaters so that currents in the heaters go through the same path but opposite direction,
suppressing the amplitude of the fields produced by the pair. The magnetometers used in this work
have two sets of pairs all wired in series, which are oriented to suppress the residual fields even
further. When only a single magnetometer is used, the dipolar field pattern expected from the
heater sets ensures BH ∼pT and are easily cancelled by the application of suitable bias fields. In
this case, DC currents can be used without any change in noise level, although there are noticeable
shifts in the magnetometer offset.
When the magnetometers are placed in an array, the field BH,ij produced at magnetometer i
by magnetometer heater j for each heater is different and depends upon the details of the spatial
57
Figure 3.5: H-bridge schematic diagram. The switches are controlled synchronously by digital signals Aand B to switch the connection polarity across the load.
configuration of the system. Additionally, dipolar patterned fields from the heaters may make the
fields BH,y,ij larger for i 6= j. Cancellation of these fields by a single tri-axial set of nulling coils
is impossible. This problem is solved either by adding coils to individually adjust the magnetic
fields in each cell, or by operating the heaters at AC frequencies fast compared with the Larmor
frequency associated with the heater fields, i.e. ωH ΩH. For the AC case, the atoms respond to
the time-averaged field, which of course is zero. Using the standard H-bridge circuit (see figure
3.5, and described in detail in appendix E), we can use a single DC source and reverse the current
through the heaters at rates of 100 kHz or more. This frequency allows the weak requirement that
the heater fields be BH 25 µT at the magnetometers to prevent heater-related response from the
magnetometer. When using the heaters at AC, we have noticed that the silicone encapsulating the
resistive heating element tends to disintegrate and leave behind its fiberglass mesh substructure.
This substructure is insulating as well, and we have not had any problem with the heaters shorting.
Chopping the current direction through the heaters introduces a noticeable amount of electronic
noise at high frequencies, which gets attenuated and aliased into the frequency band of our noise
measurement. Since the noise looks like it comes from 10 MHz ring-down of some transmission
line, this may be fixed by using purely sinusoidal heater drivers.
As described in section 3.1, the time constant for reaching thermal equilibrium for the magne-
tometer is around 40 min. This makes heating at constant output power inconveniently slow, even
when the steady-state power is known in advance. A software feedback loop is used to control the
heaters to optimally reach their set-point, using the thermistor read by a NI PXI-4065 DMM to
58
measure the temperature. When more than one magnetometer is operated at once, a digital circuit
switches which thermistor is connected to the DMM, and the software PID state is stored for the
next iteration. This feedback loop must be disabled while the FPGA is acquiring data, mainly
because the data acquisition program and heater control program compete for resources and the
currents used by the DMM to measure the resistance of the thermistors produces a magnetic field
noticeable to the magnetometers. A future design of the magnetometer should consider using ther-
mocouples to measure the temperature. Many thermocouple types utilize magnetic materials, such
as chromium or iron, but type-T thermocouples (Copper-Constantan) should be acceptable.
We have found some thermistors contain magnetic components, typically in the leads. The
thermistors used here are nonmagnetic, Honeywell 112-103FAJ-B01, 10 k NTC type. Details
about the thermistor can be found in appendix E.
3.3.2 Coils and coil drivers
Each magnetometer has a set of tri-axial square coils centered around the cell detection volume.
These coils are used for nulling the external fields to reach the most sensitive magnetometer regime,
for the application of calibration fields to characterize the magnetometers, for the application of
large magnetic fields necessary for z-mode modulation, and for feedback operation (see chapter
6). Noise in the circuitry used to drive these coils results in self-inflicted magnetic field noise
at the magnetometer, causing a corresponding decrease in SNR. Even worse, the noise at each
magnetometer is uncorrelated, eliminating suppression by difference detection. A coil driving
circuit was designed as part of this work is described in detail in appendix E. Its main purpose
is coupling an adjustable DC offset to an arbitrary applied field with the lowest possible resulting
current noise. It has a potentiometer to control the DC offset and a BNC front-panel connector
to couple any other signals onto each axis. On the lowest (experimentally usable) noise setting, it
currently has a floor of about 3 fT/√
Hz.
59
The final electrical component is the photodiode amplifier. We use SRS 570 current to voltage
converters. These devices have widely configurable gain and filtering options. Anti-aliasing filter-
ing is performed with the configurable front-panel filter options of these detectors. We typically
use 12 dB 300–1000 Hz low pass filtering for DC mode operation and sampling at 20 kHz.
In our effort to keep electronics outside the shielded room, the photodiode leads are quite long
(10 m). This is generally inadvisable [Hobbs, 2000]. We have found that in order to minimize
electronic pickup noise, some attention must be paid to the wiring itself. For low noise shielded
design with easily purchased modular components, we have selected Cat. 6 shielded twisted pair
Ethernet cable. This cable contains four pairs of twisted pair conductors, each with a different turns
ratio to minimize cross-talk and the entire cable is shielded with copper foil (some cheap cables
come with steel shielding, so each cable should be tested with a magnet). The signals on each
cable are selected to have relatively uniform range to further limit cross-coupling. For example,
the heaters are all on a single Cat. 6 cable, the photodiodes on a different one, and then one cable
each for the three sets of nulling coils and the thermistor of a single magnetometer. The cable
shields are all grounded on one side, outside the shielded room.
We have also notice 60 Hz and harmonic pickup is limited by replacing the standard AC outlet
equipment cables with shielded ones.
3.3.3 Data acquisition system
One significant part of this work was streamlining the data acquisition (DAQ) process to alle-
viate the difficulties in optimizing the system and to make the device more amenable for use with
human subjects.
The signal from a SERF magnetometer depends on many particular parameters of the exper-
iment, and hence must be calibrated with a known magnetic field in order to have an absolute
measurement of the background field amplitude, which is most important for characterizing the
noise floor of the magnetometer. The magnetometer response (2.38) changes when just about any
experimental parameter is changed. It is sensitive to all parameters of the pump and probe lasers
(intensity, frequency, and waist), the temperature of the vapor cell, and the ambient magnetic fields
60
when the measurement is performed. Further, it can be difficult to determine using a real-time
measurement whether the SNR is being improved or degraded when changes are made to the ex-
perimental parameters. One common problem is adjusting the probe power, where the photodiode
signal size increases with probe power, but the noise can either decrease or increase depending on
the probe frequency and other system parameters (see chapter 6 for a detailed analysis of this and
other noise issues).
Quick calibration mitigates these problems. The original DAQ program was written and de-
veloped as part of this work to work on LabView DAQmx devices, specifically the PCMCIA
DAQcard-6024E, and to calibrate and analyze information from a single magnetometer. This struc-
ture has been expanded to four channels and improved by Matt Kauer to operate on an FPGA-based
DAQ device (NI PXI-7851R). A block diagram of the program is shown in figure 3.6, consisting
of four main phases:
• Calibrate the frequency response of the output circuitry that drive the calibration coils
• calibrate the magnetometer frequency response, accounting for the output circuitry and fre-
quency response in the coils themselves (if any)
• take a recording without any applied applied signal to measure the noise power spectral
density (PSD)
• Have the option of reusing the same calibration to take other noise measurements
These separate steps are discussed in detail in chapter 4.
Typical sampling parameters for the measurement period are 20 kHz sampling frequency (to
avoid aliasing, we down-sample to 1 kHz during analysis) and 5-300 s sampling period. The sig-
nal is typically only conditioned with output filters on the transimpedance amplifiers described
previously.
61
Figure 3.6: Flow chart of the magnetometer characterization program. The magnetometer does not needto be recalibrated before every noise run. Generally, if nothing has been intentionally changed, the samecalibration can be reused until the DC fields have significantly changed.
62
3.4 Magnetometer array
The magnetometer described above is a complete unit, encapsulating all the necessary parts to
operate in the SERF regime. As a result, individual magnetometer channels are unconstrained with
respect to one another, due to local fiber coupled optics and individual magnetic field control at
each cell. Making an array of such units is as easy as constructing them and ensuring the operation
of one does not interfere with another. In the work described here, we have operated with an
array of four such magnetometers, and the only limits on expanding the system are in acquiring
additional auxiliary equipment needed to run the magnetometer (power supplies, data acquisition
devices, etc).
DC fields in the MSR, 10–50 nT, decrease the fundamental sensitivity according to (2.24) and
require nulling. Figure 3.7(a) shows a sweep of By and a fit with R = 400 ≈ Γ to (2.24) with
Ωx,Ωz = 0. For this R, By must be nulled to better than |By| ≤ 2 nT to remain in the sensitive
linear region of the response. If the MSR magnetic fields were sufficiently uniform, we could use a
single set of tri-axial coils with side length a and array channel baseline spacing d such that a d
to simultaneously cancel the magnetic fields at all magnetometers. Having tried this, we have
found two problems with this approach that resulted in the choice of small individual cancellation
coils for each magnetometer.
First, residual MSR magnetic field gradients are too large on the scale of the spacing of our
magnetometers to allow a uniform nulling field to be used. Figure 3.7(b) shows the response pre-
dicted by (2.24) for four magnetometers in the presence of measured MSR magnetic field gradients
and R from the fit in figure 3.7(a). Bx, By, and Bz are the differences from the measured zero val-
ues at each magnetometer channel. As shown in figure 3.7(b), these gradients cause a decrease in
both the single channel sensitivity and the effective multichannel By operating range (highlighted
in yellow), which has been decreased by a factor of 4 to |By| ≤ 0.5 nT.
Second, and most importantly, are the effective fields caused by the pump laser, described in
section 2.5. At high optical depths corresponding to high temperatures, the pump beam is absorbed
before polarizing the entire cell length unless it is detuned ∆ν1 > 1 from resonance, resulting
63
(a) Swept field response (points) and fit (line)
(b) Simulation, using the width in (a)
Figure 3.7: (a) Magnetometer signal as a function of applied field (dots) and a fit to an amplitude scaledversion of (2.24). (b) A simulation of Px for multiple magnetometers in the presence of the measuredmagnetic field gradients from the zero values at each channel. With only a single set of tri-axial coils, theseare the minimum gradients in the shielded room, and represent the best-case operating point for the fourmagnetometers. In this case, a compromise value for By must be chosen, reducing the usable operatingrange. This illustrates the need for local control over the magnetic fields at each cell.
64
Table 3.2: A summary of the various options on where to set pump parameters to operate the magne-tometer. Qualitatively, for high optical depth cells and uniform pump transverse profile, it is difficult tosimultaneously satisfy the three desired optimum conditions: R = Γ, ΩLS = 0, and pumping the entire cell.
R0/Γ ∆ν/(∆Γ/2) Polarization light shift total sensitivity
1 Low Low Low Low since atoms not polarizedHigh Low High Low Low since R
(R+Γ)2 1
1 High ΩLS
R= ∆ν/(∆Γ/2) Low since high light shift
High High
in a nonzero ΩLS. In other words, looking at pump propagation through the cell (figure 2.18),
considering the equation governing the magnetometer sensitivity ((2.35)) and the light shift as a
function of detuning (figure 2.19), there are essentially three operating regimes, outlined in table
3.2. For high optical depth cells, there is no way to simultaneously optimize R = Γ, ΩLS = 0, and
maintaining pump light throughout the cell.
The magnitude |ΩLS| is roughly the same from cell to cell, but the direction with respect to the
nulling coils depends on the orientation of the magnetometer and the direction of s. For our laser
parameters, we expect the pump light to cause effective fields of magnitude ∼ 50 nT. Orienting
all the magnetometers to have the same light shift with respect to the external coils mitigates the
problem, though the nonuniform light shift within each cell due to the spatial intensity profile of
the laser results in a broadening of the magnetometer linewidth [Appelt et al., 1999]. The light
shift still presents a problem for the prospects of differing individual channel orientation and tilt,
or nonplanar arrays. There are also practical problems associated with large coils, such as limited
magnetometer adjustment and complications with calibration fields when the magnetometers are
not aligned with the coil axis. These problems are all solved by using individual sets of coils for
each magnetometer.
The magnetometers were mounted on a plastic mounting plate which allows the adjustment
of the magnetometer baseline spacing figure 3.8. This baseplate is mounted to an existing SQUID
gantry in the MSR, which allows 6-axis adjustment of the magnetometer position. Figure 3.8(bottom)
shows a photograph of the array mounted to the gantry in the MSR. Human subjects lie on a bed
65
(a) Array schematic, from above
(b) Photo of array mounted to gantry above subject
Figure 3.8: (a) Four channel magnetometer array, with 7 cm channel spacing. This array is mounted onan existing SQUID gantry top-down, as in the photo (b). In adult subject trials, the subject lays on a bedand the array is centered over the subject’s heart by equalizing the signal in all four channels in real time.For fMCG studies, the array is angled and positioned below the naval. The x, y, and z axes in the textrefer to magnetometer body axes. Here, all y are aligned and correspond to the sensitive direction for allmagnetometers.
66
under the array, which can be translated, rotated, and tilted using the gantry controls. For conve-
nience in these initial tests, our measurements were made with planar square array and channel
spacing of 7 cm. Physical limitations permit a 4.5 cm minimum planar array spacing for all four
elements. More magnetometers can be added, with 7 cm spacing in the vertical direction of figure
3.8(a), while the optics tubes require the spacing in the horizontal direction to be >10 cm. In our
setup, the four elements operate independently and could be tilted or translated with respect to
one another by adjusting their mounting to the plate. It should be straightforward to implement
non-planar geometries.
Using three layers of 2 mm thick thermal insulation and including the distance from the front of
the cell to the position of the center of the intersection of the pump and probe beams, the minimum
physical distance to the skin for each magnetometer in this design is about 1 cm. When using
the magnetometers, we typically avoided putting them in direct contact with the skin, as normal
breathing motion tended to cause the subject to bump the device and produce vibration noise. The
safety margin was on the order of an additional 1 cm spacing from the skin (2 cm total), somewhat
larger than Knappe et al. [2010], where where the reported distance was 5 mm from the skin. Better
engineering would allow the 1 cm minimum distance to be used in measurements.
67
Chapter 4
Data Collection and Analysis
This chapter describes the characterization and operation of the magnetometer, as well as com-
monly used signal processing techniques for the application of biomagnetism. After the previous
chapters on the basic theory and apparatus design, this chapter serves as something of a manual for
the actual use and characterization of the magnetometer.
The initial setup of the magnetometers includes alignment of the laser paths, initial balancing
of the polarimeter, and residual DC field nulling. After this procedure, the magnetometers should
be as close as possible to the zero field regime, and at their most sensitive given other experimental
parameters. Then each channel’s response, both amplitude and phase, are calibrated against known
applied magnetic fields.
After the calibration phase, the magnetometers are ready to measure fields of interest. For
characterization of magnetometer performance, these are the background fields in the MSR, while
for biomagnetism these will be MCG or fMCG.
4.1 Setting up magnetometers to operate in DC mode
For biomagnetism studies, the array is mounted on the SQUID gantry. Each magnetometer
is aligned separately. The only flexibility in the pump beam path is in the relative orientation of
68
the optics tube to the cell holder, so the first step is aligning the pump beam with the cell. We
have found it sufficient to align the pump beam to travel through the center of the front and back
windows.
The probe beam alignment is performed using the mirrors (see figure 3.1 and figure 3.2) with
the orientation of the optics tube fixed. As described in section 3.2, the mirror mounts allow mirrors
to be rotated about their center and, using one of two screw holes, tilted back and forth. One of the
tilt screws should be used for each mirror mount, even if it is not necessary, to minimize vibrations
that would occur in the otherwise unconstrained flexure hinge assembly.
Once the beam paths are aligned, the polarimeter needs to be balanced. This is done by block-
ing the pump beam (to ensure Px = 0), and measuring the output of the transimpedance amplifier.
The half-waveplate in figure 3.3 is used to zero the output signal (balance the intensities on the dif-
ferential photodiodes). Sometimes this probe-only signal will be unstable, with larger than normal
low-frequency noise in the signal. This indicates either a gross misalignment of the polarization
maintaining fiber such that the output polarization drift is large (see section 3.2), or more likely
an increased sensitivity to the alignment of the probe beam, which varies slowly with stray air
currents and temperature fluctuations of the system.
In the latter case, the increased sensitivity can often be removed by more careful probe beam
alignment, ensuring the beam path traverses through the center of the optics rather than too near
their edges, and that the light incident upon the photodiodes is centered on the photodiodes and
does not overfill them. The same symptoms occur when the probe intensity is too high at the photo-
diodes, presumably causing non-common mode nonlinear response in the photodiodes themselves.
This can be simply tested by decreasing the probe intensity and rechecking the noise spectrum.
Finally, we have been able to decrease this low frequency type noise by measuring photodiode
response vs. intensity curves and choosing uniform pairs to make the balanced polarimeter. This
process is labor intensive and provides marginally better than random matching. In the future,
photodiode pairs constructed on a single chip should be considered to minimize differences due to
manufacturing processes.
69
4.2 Residual field nulling
The nulling process is similar to previous DC mode SERF magnetometers [Li, 2006, Seltzer
and Romalis, 2004] and makes use of terms proportional to orthogonal fields in the steady-state
response (2.24), which we reprint here for convenience,
Px = Pz
(Γ′Ωy + ΩxΩz
Γ′2 + Ω2z
)≈ Pz
Ωy
Γ′.
From the equation above it is obvious that the magnetometer will respond with the application of
a field Ωx only if Ωz 6= 0, and vice-versa. We find the DC null fields for a single magnetometer as
follows:
• sweep By and set the operating point in the center of the dispersive curve (figure 2.12a),
where the magnetometer is most sensitive.
• apply a ∼50 pT oscillating field B0 sin(ωt)x at 10–30 Hz and adjust Bz until there is no
response to B0, ensuring By remains at its most sensitive operating point. If the response is
difficult to find at first, a larger field may be used.
• apply B0 sin(ωt)z and adjust Bx to null the response, again ensuring By is adjusted to main-
tain maximum sensitivity.
• repeat until residual response to Bx and Bz is minimized.
For the modular four-magnetometer array, we repeat this procedure for each individual unit. It-
erating the procedure two times eliminates any offsets of the local field at the first magnetometer
caused by adjusting the fields in the later ones to below the adjustment procedure sensitivity. The
easy extension of the single magnetometer nulling procedure to all four magnetometers is one
important feature of the modular array.
There is no cross-talk between our four channels because the magnetometers are fundamentally
passive devices. Non-uniform light shifts from the Gaussian pump laser intensity profile cause
non-uniform effective magnetic fields as in (2.56). Due to the residual Ωz term in (2.24), each
magnetometer retains some sensitivity to Bx. This is typically a correction of ≤ 10% and its effect
is ignored when characterizing the magnetometers.
70
As discussed in section 3.4, the magnetometers must be within approximately 1 nT of their
null point in order to maintain uncompromised sensitivity. Background field variations in the
shielded room are typically smaller by an order of magnitude or more on the timescales of of a
30 s MCG measurement. Long-term variations can take the magnetometers outside of the optimal
measurement range, but are easily compensated between measurements. Very occasionally, the
DC offset of the fields in the MSR will shift a large amount on an inconvenient timescale (i.e once
per few minutes), which essentially prohibits the use of the magnetometer. The source of these
fields is probably the construction of a building physically near the MSR, including slow moving
cranes, large trucks, and industrial elevators. If the changes are only in the detection direction, they
can also be compensated by negative feedback of the magnetometer signal to the magnetic coils,
which we will discuss in detail in section 6.1.2.
4.3 Characterization procedure using DAQ program
Once the DC null fields have been set, each magnetometer is simultaneously characterized
using the DAQ program. The normal characterization and data acquisition stages described below
all use sampling frequency fs = 20 kHz, with a two-pole low-pass filter set between 300–1000Hz,
using a 16-bit FPGA digitizer.
4.3.1 Coil driving circuit calibration
Digital to analog converter (DAC) outputs on the FPGA board allow the DAQ program to send
signals to the magnetometers. However, these signals need to be conditioned and converted into
properly sized current sources to drive the magnetic coils. The coil control electronics box (see
appendix E) has a control input for this purpose. The input voltage is scaled, added to a DC offset
set by a front-panel potentiometer, and applied to a voltage to current converter subcircuit, which
produces a constant voltage across an output resistor R0. If we take the input voltage to the coil
control box produced by the DAQ system as Vi and the output voltage across the output resistor as
Vo, then the complex gain of the output circuit
71
Table 4.1: A list of frequencies, in Hz, used to calibrate the magnetometer frequency response. 8 cycles ofeach frequency are applied to the device under test and the response detected.
2.57 5.45 18.7 29.3 37.5 37.5 53.2 69.7 75.1 83.391.9 101.3 113.7 129.1 134 153.7 167.7 189.3 197.5
A(ω) =Vo(ω)
Vi(4.1)
is characterized by applying a known, constant amplitude sinusoidal signal Vi(t) = Vi sin(ωt)
at frequencies across the 200 Hz bandwidth of interest and recording Vo(ω). Vi is applied for a
length of time T ≥ 8× 2πω
, and then the resulting signal fitted to a sinusoid to extract the response
amplitude and phase Vo(ω). Knowing the output resistance across which we measured the voltage
and the magnetic field coil calibration χ(ω) allow us to back out what magnetic field B0 is applied
to the magnetometer by using the output of the DAQ system and the coil control box,
B0(ω) = A(ω)χ(ω)ViR0
= η(ω)Vi. (4.2)
Table 4.1 shows a table of all the frequencies used to characterize the magnetometer. The coil
driving circuitry needs to be calibrated whenever the gain of the box is changed. In the absence
of any changes, the DAQ program will calibrate the driving box once per day to account for slow
circuit changes due to temperature and aging, and as a consistency check. Note that χ(ω) is
constant for the small coils wrapped around the magnetometers because their small size and spatial
separation from any mu-metal ensures their inductive reactance is low compared to the coil-driving
circuit output impedance at the frequencies of interest.
4.3.2 Magnetometer calibration and initial noise measurement
After A(ω) is determined, the magnetometers themselves need to be calibrated. The DAQ
output remains connected to the coil control box (as above) but the DAQ input now reads the
output voltage from the transimpedance amplifier (Vm). This frequency response is taken in the
same way as described above to characterize the magnetometer response to an applied field,
ξ(ω) =Vm(ω)
B0
, (4.3)
72
which can be rewritten using (4.2) as
ξ(ω) =Vm(ω)
η(ω)Vi. (4.4)
The technique described above determines ξ(ωi) using (4.4) at the frequency points listed in table
4.1. Then ξ(ω) is extrapolated by fitting this function to an empirically modified version of the
theoretically expected low field magnetometer frequency response (2.36),
|ξ(ω)| = a
√Γ′/2√
(ω − ω0)2 + (Γ′/2)2×(1 + bω + cω2
)(4.5)
arg(ξ(ω)) =180
πtan−1
(ω − ω0
Γ′
)(4.6)
where the last term in parenthesis has small b and c coefficients to account for phenomenological
differences between (2.36) and the actual measured curve, likely resulting from nonuniformities in
the cell polarization and pump intensities.
The results of one such calibration are presented in figure 4.1(inset). At this point, the DAQ
program measures the output of the magnetometer without any applied signal for several seconds,
and uses (4.3) to deconvolve the unknown applied magnetic field which produces the measurement,
Vm. The field spectral density (√
PSD) is characterized to display the noise floor and identify any
noise peaks usingδB√Hz
=
√PSD(Vm)
|ξ(ω)|. (4.7)
Figure 4.1 is an example of the result of the calibration procedure described above. The points
correspond to data taken at the various frequencies, and fits with 95% confidence intervals are
shown overlaid on top, which allows for a 10% variation in the fit amplitude and uncertainty
in the precise form of the frequency response, a problem for producing high quality difference
signals. These measurements are normalized to the maximum in each measurement, but the overall
response amplitude is different for each magnetometer.
4.3.3 Additional noise measurements (Probe and electronic)
The calibration procedure described in section 4.3.2 provides a measure of the total signal from
the output of a magnetometer and treats it as if it were all generated by magnetic field. A perfect
73
Figure 4.1: An example of the results of the amplitude and phase calibration. The blue points are the resultsof measurements of responses to the system at different frequencies (normalized). The solid black lines arefits to a modified theoretical model, and the red dashed lines indicate the 95% confidence interval for thefit model. The amplitude and phase fit parameters are not required to be consistent. The shapes of thecurves for each channel are typical (e.g. channel 4 is always broad, channel 3 often has a nonzero centralfrequency). We are still unclear what is different in each cell to cause these discrepancies.
74
SERF magnetometer would be limited by quantum noise, and it is useful to estimate the noise from
non-magnetic sources.
According to (2.24), blocking the pump beam and measuring the noise leaves the magnetome-
ter insensitive to magnetic fields in any direction (Pz = 0), and the magnetometer signal is a
combination of fundamental quantum noise (spin-projection and photon shot noise) as well as
technical noise associated with electrical pickup and probe fluctuations kind. Treating this signal
as an effective magnetic field (scaling it by ξ(ω)) gives an estimate of the total non-magnetic noise
in our system.
The argument above disregards two subtle signal sources that are sometimes important. First,
noise induced by the pump beam will clearly not be detected when the pump beam is blocked.
Several individual avenues are available for the pump beam to cause a change in signal: direct
detection of scattered pump light by the photodiodes, changes in pump intensity or frequency
causing changes in 〈Pz〉 averaged over the probe beam path, and changes in the pump intensity
and frequency changing the effective field induced by the light shift (2.56), which couples time
varying fields in the probe direction to Px with time varying strength.
The purely electrical component of the noise of a magnetometer can be measured by blocking
the pump and probe beams. We do not see any difference in the noise spectrum (although there is a
small offset shift) between blocking and unblocking the pump, suggesting stray pump light itself is
not an important factor. We will address two issues relating to changes in pump laser parameters in
detail (chapter 5), but they can be large because of nonlinear absorption effects in the optically thick
vapor cell. This type of noise manifests itself in differences between the magnetic noise floor of
two channels, when not limited by probe noise. Thus, blocking the pump beam can underestimate
the technical noise of the magnetometer in the case where the technical noise is induced by the
pump itself.
The second subtle signal source is residual pumping by the probe beam itself. This can be
caused by some remaining elliptical polarization in the probe beam. Considering the extinction
ratio of the polarizers used for the probe (> 10, 000 : 1), the most likely source of elliptic polariza-
tion is birefringence in the cell glass, perhaps induced through stress by physically clamping the
75
cells in the Boron-Nitride. Here, the atoms are weakly pumped to a polarization Px determined in
part by the fluctuating ambient field amplitudes. This effect causes the probe noise to over-estimate
the non-magnetic noise level, since it adds back in magnetic noise. This effect is non-negligible,
especially in channel 3, where the magnetometer retains obvious sensitivity to fields perpendicular
to the normal direction DC-mode direction. This may be due to the higher pressure-broadened
linewidth leading to higher probe absorption rates in this cell.
With these two caveats, we take the probe noise as a measure of the total non-magnetic noise
of our system. Figure 4.2 shows the result of the calibration process, with both normal magnetic
and probe noise. For the low-frequency components of the cell, the probe noise dominates, while
the noise floor is dominated by other sources. The noise floor is constant over time, but residual
sinusoidal magnetic fields often dominate the signal. Figure 4.2 shows a measurement with a rel-
atively quiet background, displaying only a few such peaks near at 6.7, 50.5, 60, 101, and 120 Hz.
The 6.7 Hz peak in particular is the largest background signal in the room, is well-characterized
by a sinusoid, and is circularly polarized. We have verified that it emanates from a large HVAC
fan located almost directly above the MSR on the next floor, perhaps 5–10 m from the center of
the MSR. Though this makes real time analysis of the magnetometer signals difficult, large si-
nusoidal signals tend not to interfere with MCG measurement because they are narrowband and
easily identified. Standard signal processing techniques can easily remove them (see section 7.2).
An important and often overlooked feature of figure 4.2 and figure 4.1 pertains to bandwidth.
Looking at figure 4.1, one is tempted to conclude that the bandwidth of the magnetometers may
be a problem for biomagnetic signals that require measurement of frequency content up to 100 Hz
or more. If the signals were limited by the noise performance of the magnetometer alone, then
this would be true. Inspection of figure 4.2 does not show a decrease in performance at the knee
frequency of the calibration. The reason is because the magnetometers are dominated by magnetic
noise, which is white. A more meaningful characterization of “bandwidth” would be the frequency
at which the non-magnetic noise level of the magnetometer is equal to the magnetic noise level as
the sensitivity of the magnetometer decreases. We can reasonably conclude from a measurement
76
Figure 4.2: Calibrated noise measurements for the four channels (ch1 on top to ch4 on bottom), on arelatively quiet day. The 6.7 Hz peak from an HVAC fan is easily identifiable as the biggest signal inthe MSR. This plot illustrates the typical relationship between magnetic noise and probe noise for eachmagnetometer (e.g. the probe noise in channel two is typically a larger fraction of the total noise than theother channels).
77
like figure 4.2 that the usable bandwidth of our atomic magnetometer in this particular MSR en-
vironment is over 200 Hz. Said differently, judging by the noise floor, one of our channels will
perform as well as any other magnetic detector in the same shielded room to frequencies past
200 Hz, even if the other sensor is a SQUID magnetometer with a much higher bandwidth.
4.3.4 Chirp response measurement
Finally, the frequency calibration described above can be generalized to any swept-frequency
measurement. For example, if we generalize the input amplitude for every sinusoidal frequency
to a time-varying function, the response to that function, Vi → Vi(ω(t)) in (4.1) and (4.2), then
the response to that function can be characterized by deconvolution (which is a generalization of
matching amplitudes and phases of discrete sinusoidal signals).
We have recently switched to this process, using a frequency chirp signal applied to the mag-
netometers, and measuring the magnetometer response this way. The benefits to this technique are
a faster calibration with better resolution between frequency points. Chirps measuring the same re-
sponse take only a fraction of the time, mainly because the sinusoidal technique requires 8 cycles,
which can take seconds for the lowest calibration frequencies. The disadvantages are that highly
spectrally peaked noise components are more likely to interfere with this technique, since a chirp
response continuously measures a signal.
The chirp response function used for the calibration is
Vi(t) = Vi sin(AteBt), (4.8)
where A sets the base frequency scale and B sets the sweep velocity. A Labview VI (Frequency
Response Function.vi) takes the applied chirp signal Bi and the response signal Vm and measures
the frequency response function
ξ(ω) =Sim(ω)
Sii(ω)(4.9)
where Sim is the cross spectral density of the stimulus signal with the appropriate conversion from
output voltage to magnetic field given by (4.2) and the response, and Sii =PSD of the stimulus
signal. To help reject noise sources, four of these data sets are averaged for each magnetometer
78
-2
0
2
Res
pons
e (a
rb.)
0.150.100.050.00 Time (s)
Meausred response Applied signal
Figure 4.3: An example of the chirp calibration signal applied (black) to the magnetometers and the re-sponse (red). Using the chirp signal allows the magnetometers to be characterized more quickly.
to calculate the frequency response function. Figure 4.3 shows an example of the time series
calibration signal represented by (4.8), and the magnetometer response from such a calibration.
4.4 Optimization
After characterizing the pump, probe, and electronic noise levels, we have a picture of how the
magnetometers are performing. Table 4.2 gives a summary of the different types of measurements
we make to characterize the limits the magnetometer noise level. We are satisfied when the mag-
netometer noise level is consistent with the expected background magnetic noise level of the MSR,
which is around 4 fT/√
Hz including the noise from our current controllers (see chapter 5). This
also requires the probe noise level to be significant below the magnetic noise level, important for
accurate single channel measurements, but even more so for multichannel measurements where
the noise suppression will depend upon the degree of correlation between the noise levels at each
magnetometer, as will be discussed in more detail in chapter 7.
Often, these conditions are only partially met. For example, the probe noise might dominate the
spectrum in the 1/f region of the spectrum, and then magnetic noise dominates, like in figure 4.2.
Alternatively, it may appear the probe noise is limiting the performance when the electronic noise
79
Table 4.2: Different types of noise measurements used to characterize magnetometers. This characteriza-tion provides a feel for how the magnetometers are performing, and what is limiting them.
Measurement Method Noise level interpretation
Magnetic Noise Nothing blockedSame as probe: limited by probe noiseHigher than probe: If channels are same, magnetic noise
If channels are different, may be pump noise
Probe Noise Block pump beam Quantum noise plus non-pump technical noise and electric noise
Electronic noise Pump and probe blocked Noise purely from electronics (mostly pickup)
is too high. We briefly discuss the process of laser optimization in order to achieve the best noise
performance, given information about the relative amplitudes of the three noise measurements.
First, if the noise level is dominated by electric noise, that means either the electronic noise
level is too high, or conversion factor from field to output current by the magnetometer is too low.
Sometimes the electric noise is too high because of a poor choice of settings on the I-V converters,
such as forgetting to apply the anti-aliasing filters or using gain that is too low (500 nA/V is
typical). Otherwise, we have worked pretty hard to eliminate electronic noise, so the easiest thing
is typically to find a way to increase the magnetometer response, in terms of absolute signal.
Considering the difference in the performance of the magnetometers, it is best to optimize
all channels simultaneously. A small oscillating magnetic field By, about the size of a normal
calibration field and near 10 Hz, is applied to all magnetometers through a large coil. The amplitude
of the response is marked, and then the pump laser frequency and intensity is tuned to increase or
decrease the response. Once a maximum is found, the fields (especially Bz) must be re-tuned, and
the process can be repeated. Watching all signals simultaneously ensures that, due to nonuniform
temperature and relaxation in each cell, all cells respond adequately. This seems to be especially
important for channel 2, where the response is always low compared with the others to begin with.
If adjustment of the pump beam alone does not fix the problem, the probe beam can be adjusted as
well. When electronic noise is most important, increasing the signal size is paramount.
If the probe noise is the largest noise source, the problem can be more subtle. For example, if
the probe noise has mostly 1/f character, this might be the result of photodiode nonlinear response,
coupled with slow variations in the exact position of the probe incident upon the photodetectors.
80
Figure 4.4: A plot of the probe noise for three different probe intensities. The highest was taken withoutany attenuation. In the other two, ND=0.3 and ND=0.3+0.5 filters were used to attenuate the probe lightbefore the fiber splitter. Though the overall signal decreases when the light is attenuated, the 1/f amplitudealso decreases. In the higher frequency ranges, electronic noise begins to be important where the bandwidthof the magnetometer decreases the response below the electronic noise threshold.
In this instance, decreasing the probe intensity (and so the response to a uniform applied field)
can decrease the noise. As an example, figure 4.4 shows three different measurements for the
probe noise under identical circumstance, except for probe intensity. The lowest intensity probe
shows better 1/f noise performance but pays a price in the higher frequency noise level. The high
frequency performance is compromised because the electronic noise overwhelms the probe signal
at lower frequency for the same bandwidth, as we discussed in section 4.3.3.
4.5 Signal measurements
Adult signal measurements are performed similarly to noise measurements. After the mag-
netometer characterization has been performed, recordings of the ambient magnetic field can be
taken with arbitrary length for 20 kHz sampling frequency. For biomagnetic signals, the longest
continuous stretch required is 300 s (for fMCG measurements).
There are a couple of techniques we use when acquiring biomagnetic signals to minimize the
time a subject must spend in the MSR. First, before the subjects arrives, the magnetometers are
mounted on the SQUID gantry and put into position above the subject bed. Magnetic gradients in
the MSR will cause the nulling fields to change as the magnetometers are physically repositioned,
81
especially for rotations which change the projection of the magnetometer plane onto the MSR
floor/ceiling plane. The magnetometers are positioned near their anticipated operating point to
minimize the required adjustments. Second, the pneumatic bed is lowered so that the subject can
slide comfortably underneath the magnetometers and into position. Then the bed can be raised
to the correct position, and finally the magnetometer can be moved into precise position by an
operator who sits inside the MSR.
For fetal MCG measurements, an ultrasound is performed to identify the position of the fetal
heart. In the initial measurements presented in this thesis, a commercial SQUID device was used
to measure the fMCG signal after the ultrasound to cross-validate SERF measurements and to
help identify an initial orientation for the SERF array. Finally, the SERF array is placed as close
as possible to the subject’s skin without the subject bumping the magnetometer during normal
respiration. The channel position for the fMCG signals in this work is shown in figure 4.5.
Background signals and the mother’s MCG can swamp the fMCG signal in real time. While
signal processing techniques do a very good job, it is important to position the magnetometer
to optimize the fMCG signal, requiring a real time or quasi-real time measurement in which the
fMCG is at least visible. We mentioned previously one large source of interference the background
spectrum, the 6.7 Hz signal from an HVAC fan. The fan can be turned off for short periods of time
(∼1 Hr) and the few fMCG measurements performed here had the fan turned off for a large part of
the measurement period.
82
Figure 4.5: An artist’s rendition of the channel positions of the magnetometers for the fMCG measure-ments. The relative position of the fetus is unknown, but the relative channel positions to the mother’s heartare qualitatively correct. From this figure, we expect the mother’s MCG signal to be largest in channels1,2, and smaller in channels 3,4, and for those pairs to have approximately equal amplitudes. All channelsshould have similar phase.
83
Chapter 5
Noise analysis
This chapter presents a careful noise analysis, both of the atomic magnetometer itself and
its environment. This noise analysis informs choices about extensions to DC-mode of operation
(chapter 6), as well as our discussion of biomagnetism and which signal processing techniques
promise to enhance the SNR of fMCG (chapter 7).
We can lump the noise sources into three categories: Magnetic, fundamental quantum, and
technical noise. Since we are interested in the application where a single magnetic source is the
signal of interest, magnetic noise includes any other magnetic source. Fundamental quantum noise
essentially limits the sensitivity of the SERF detector itself, and comes from random projection
noise in the measurement of the atomic spin and the measurement of the probe polarization rota-
tion. Technical noise is anything else that could result in spurious signal, including but not limited
to electronic pickup noise on the photodiode leads, slow rotations in the output polarization from
the laser fiber optics, and movement of the probe laser on the photodiode surface area coupled to
position-dependent photodiode response.
Table 5.1 lists the various sources of noise for each of the three types we consider for the DC-
mode magnetometer discussed so far, as well as an estimate for the noise magnitude. The rest of
this chapter will deal with each noise source individually.
84
Table 5.1: Magnetometer Noise Budget. Two results are quoted for the photon shot noise. The basic modelis (5.18) where Pz , Γpr, and Γ are all averages over the laser interaction volume. The detailed modelsare refinements, where these quantities are made position dependent and effects such as light shifts areincorporated. Where the source is not expected to produce a white frequency profile, we note the expectedfrequency profile, though in most cases these are qualitative descriptions. When the noise source can bequantitatively estimated, a value or range of values is quoted. When upper limits are placed based on ourmeasurements, these limits are quoted.
Type Source Level ( fT/√
Hz) Spectrum
Mag
netic
MSR Johnson noise 1.1Electrical current noise in nulling coils 3.987Rb Johnson noise 0.9Copper in nulling coils 0.1Subtotal magnetic white noise 4.2Fields from outside MSR (HVAC fan) (10× 103) sinusoidalFields from outside MSR(Construction) (1− 100× 103) offsetNoise from DC cell heaters ( 5)
Qua
ntum
Spin-projection noise 0.8Spin-projection noise (min) 0.5Photon shot noise (averaged model) 0.1Photon shot noise (3D model, no ΩLS) 0.2Photon shot noise (3D model) 0.7Photon shot noise (3D model, ΩLS = 0) 1.1Photon shot noise (measured) 1.5Quantum subtotal (spn and exp. psn) 1.7
Tech
nica
l
Electronic 0.5Signal aliasing suppressed -30 dB-6 dB/OctaveFiber polarization rotation δB/B < 1/500 (1/f)Probe frequency noise δB/B < 1/100 (1/f)Pump frequency noise δ B/B < δν2 (104 MHz)−1
Pump intensity noise δB/B = δP2/P2
Laser angular misalignment (P dependence) δθ δP1
P1
(50 pTdegree
)Laser angular misalignment (δ ν dependence) δθδν1 ×
(1 fT
MHz degree
)Total (no current noise in nulling coils) 2.4Total (white noise level) 4.6
85
5.1 Magnetic noise
Magnetic noise generally includes any magnetic source that falls within the frequency band of
interest and limits the achievable SNR of the desired signal. Clear examples can be seen in figure
4.2(a). An important contribution to the single channel magnetometer noise floor in figure 4.1(b)
and figure 4.2 is Johnson noise in nearby electrical conductors. These conductors carry thermal
currents which generate roughly white-noise profile magnetic fields. The most important noise
sources in our case are likely the mu-metal walls of the MSR and 87Rb thin-film remaining on the
Pyrex cell walls. Noise calculations for useful conductor geometries are available, and we use the
results of [Lee and Romalis, 2008] in our Johnson noise estimates provided below.
5.1.1 Johnson noise from nearby conductors
We characterize the MSR as a set of 6 high-permeability infinite planes of distance a = 1 m
away from the detector. The noise spectral density δB from one of these planes is
δB =1√6π
µ0
√kBTσd
a, (5.1)
in the direction of the plane area, where T is the temperature of the wall, d is its thickness, and σ
is the electrical conductivity. The noise in each direction is then the quadrature-added sum of two
planes, δBMSR = 1.1 fT/√
Hz.
There is also a film of 87Rb coating the cell. Modeling this film as a thin circular disk of
thickness d = 10 nm, radius r = 1 cm, and a = 0.5 cm from the measurement volume, the noise
contribution from this film is approximately [Griffith et al., 2010, Lee and Romalis, 2008]
δBRb =1√8π
µ0
√kBTσRbd
a
1
1 + a2/r2, (5.2)
and the noise from the rubidium is approximately 0.9 fT/√
Hz.
The next closest conductors are the nulling coils wrapped around the magnetometers, seen in
figure 3.1. The equation for the Johnson noise field far away from a conducting wire is [Lee and
Romalis, 2008]
δBwire =
√3
128µ0
√kBTσcuy
20a−5/2 (5.3)
86
where a = 2.5 cm is the distance from the wire to the measurement volume and y0 = 160 µm is
the radius of the 28-gauge wire used for the coils. Our coils consist of 2-sets of four such physical
wires for each axis, or 32 total wires. Up to factors of order unity having to do with the vector
nature of the fields and whether they are correlated for wires that are connected, adding the fields
in quadrature yields a modest δBcoils = 0.14 fT/√
Hz.
5.1.2 Magnetic noise from nearby electronic circuits
There are four electrical signals that carry currents near our sensor volume as a result of the
heaters, thermistor measurement, photodiodes, and the magnetic nulling coils themselves. The
heaters operate at around 0.5 A current which would be problematic, but is alleviated by the tech-
niques described in section 3.3 and appendix E. Noise measurements of the type in figure 4.2 do
not show any difference between the heaters on or off, giving an upper bound for the noise due to
heater currents. The photodiode currents are very small (∼ 1 µA), and the thermistor measurement
current is not applied during magnetic field measurement.
That just leaves excess current noise in the circuit which drives the nulling coils. We measured
the electrical current noise in the circuitry used to drive our nulling coils using a transimpedance
amplifier. This contributes ∼4 fT/√
Hz to the magnetic noise level, depending on the coil current
controller settings (see appendix E).
5.1.3 Noise from sources outside the MSR
Figure 4.2 shows that attenuated sources outside the MSR can still be seen in the magnetic
noise level. This is especially true for low frequency noise sources, since the overall MSR shield-
ing factor decreases as the eddy-current shielding factor decreases. One large source, mentioned
previously, is the HVAC fan, which has a consistent ∼ 10 pT amplitude signal.
As a guess of the magnitude of the problem, we take a dipole model of an automobile, from
Vrba [2002, fig. 2]. This figure can be used to find (very roughly) the field from a car as a function
of distance, shown in figure 5.1. Due to the construction, trucks, cranes, and construction elevators
are all located ∼50 m from the MSR. If these fields have a similar or larger field than a car, this
87
Figure 5.1: Field from a dipole model of a car [Vrba, 2002]. The dashed line is the actual field, the solid lineis the field attenuated by a factor of 250, the approximate DC attenuation of the MSR. It seems reasonableto suspect fields from larger objects like trucks, cranes, and elevators to have similar or even larger fields.The proximity of heavy construction near the MSR during the measurement period of the data representedhere causes potentially large fields compared to the background noise level.
could cause real problems in recording measurements (since there are large changes in the field
amplitudes). It helps that the characteristic frequency is very low ( 1 Hz).
5.2 Fundamental quantum noise sources
Though single channel SERF magnetometers are often limited by environmental magnetic
noise, noise sources below this level are still important for signal processing applications, and
fundamental noise limits impose a hard lower bound on these capabilities. There are two fun-
damental quantum noise sources which limit the maximum theoretical magnetic field resolution:
spin-projection noise and photon shot noise. These parameters depend on the details of the laser
parameters, ambient magnetic fields, cell properties, magnetic field gradients, etc.
A brief summary of the relevant calculations is presented in the appendix of Ledbetter et al.
[2008], for the basic case of uniform laser parameters across the cell measurement volume, and
no light shifts. We introduce those results here, before expanding upon this basic theory, adding
88
additional refinements to obtain a more realistic picture of these noise sources in our system and
how they might be optimized.
5.2.1 Spin-projection noise
Spin-projection noise comes from using the probe beam to measure the polarization Px per-
pendicular to the quantization axis, defined by the pump laser. The uncertainty in the angular
momentum Fx per atom is
∆Fx =
√〈F 2
x 〉 − 〈Fx〉2 (5.4)
Assuming no magnetic fields and no ellipticity to the probe polarization, 〈Fx〉 = 0 by symmetry.
The remaining term 〈F 2x 〉 = Tr(ρF 2
x ), can be evaluated in spin temperature as Tr(ρF 2x ) = q(P )/4,
for normalized Tr(ρ) = 1 [Ledbetter et al., 2008]. Averaging the measurement from a set of N
uncorrelated atoms, the uncertainty in the total angular momentum vector decreases according to
Poisson statistics,
δFx(P ) =
√q(P )
4N. (5.5)
For a continuous measurement and the measurement time long compared to the spin-decoherence
time [Kornack, 2005, Ledbetter et al., 2008],
〈δFx〉 = δFx
√2T2
t, (5.6)
where the angular momentum coherence time is T2 = q(P )/Γ′ (chapter 2). Finally, assuming
the spin-temperature distribution allows us to relate the total angular momentum and electron spin
through the slowing down factor, 〈Fx〉 = q(P ) 〈Sx〉. With Px = 2 〈Sx〉 as usual, we can use (5.6),
(5.5) to find
δPx =
√2
NΓ′t. (5.7)
In our geometry, δPx mimics a magnetic field, and can be expressed as an effective noise field
using (2.24) and (2.25) as
δBy = B0
√2
NΓ′(5.8)
δBy ≈Γ′
Rγ
√2Γ′
N(5.9)
89
where the approximation holds for small fields, and B0 relates to the prefactor of (2.24),
Px =By
B0
(5.10)
B0 =
(Pz
γΓ′
Γ′2 + Ω2z
)−1
(5.11)
Pz =R
Γ′(5.12)
NoteB0 has units of magnetic field, and is minimized for the normal optimal magnetometer scaling
response, i.e. for low fields and R = Γ. We have assumed Ω2x + Ω2
y Ω2z in the expression for
By. Equation (5.11) may look strange, since it could be further simplified if the expression for Pz
in terms of R and Γ is used. However, the form of (5.11) will be useful shortly, when diffusion
becomes important. When calculating Pz, we will handle Γ′ differently than in the other terms in
B0.
For small fields, spin-projection noise can be minimized forR = 2Γ, which is a larger pumping
rate than the condition for optimizing the signal size (2.26) (where R = Γ). The minimum spin-
projection noise is [Ledbetter et al., 2008]
δBmin =3
γ
√3Γ
2N. (5.13)
Assuming the number of atomic participating in the measurement are N = nV , and the effec-
tive volume V = πw22 × 2w1 is the intersection of the pump and probe beams, and using the
experimental parameters in appendix B, we estimate the spin-projection noise to be 0.8 fT/√
Hz.
The minimum relaxation rate at our densities and accounting for relaxation due to diffusion is
0.5 fT/√
Hz.
5.2.2 Photon shot noise (basic)
Photon shot noise occurs because the polarization of the probe laser is determined by counting
the difference in the number of photons arriving at the two different photodiodes in figure 3.3.
Counting uncorrelated photons results in uncertainty per unit frequency in the polarization of the
90
probe beam as δφpsn ∝ N−1/2ph . The number flux of photons that participate in the measurement is
δφpsn =1
2√Nph
(5.14)
Nph =P2
hν2
e−nlσ02(∆ν2), (5.15)
where N is the photon detection rate for the probe power P2, which is measured at the front of the
cell. The exponential term accounts for probe absorption by the atomic vapor.
Since our signal is a polarization rotation, shot noise in this signal mimics a magnetic field, and
once again (2.24), (2.25) and now (2.38) can be used to relate the uncertainty in the rotation to an
uncertainty per root frequency in magnetic field (neglecting hyperfine structure) [Ledbetter et al.,
2008]. To make extensions to this description easier in the following sections, we express relation
between the effective magnetic field noise and photon shot noise through a calibration factor, the
polarization rotation per unit field,
δφ =φ
By
By (5.16)
φ
By
=1
4refcnlD(∆ν2)
1
B0
. (5.17)
We have assumed uniform response from all the atoms in the probe beam area and along its prop-
agation length through the atomic medium.
The full expression for the photon shot noise is then
δBy = B02√
ΓprOD02nlA∆ν2D(∆ν2), (5.18)
with OD02 = nlσ02(0) the D2 optical depth on resonance. We recover Ledbetter et al. [2008, eq.
A10] in the limit ∆ν ∆Γ/2, and assuming the probe beam area is spread uniformly across
the square cell. Using the spatially averaged laser intensity for the parameters in appendix B to
evaluate (5.18) suggests a photon shot noise level of 0.1 fT/√
Hz.
5.2.3 Experimental measurement of fundamental noise
Comparison with the probe noise measurement described in section 4.3.3 is not as straight-
forward as one would like, because it includes electronic and other technical noise sources. We
91
(a) δB vs ∆I∆B for P0 = 1 mW (b) ∆I
∆B vs P0 to maintain δB=1 fT/√
Hz
Figure 5.2: Plots of the electrical measure of photon shot noise as various experimental parameters arevaried. P0 is not independent of the photon shot noise, so while (b) appears to suggest lower power isbetter, there is a hidden linear dependence on P0 since ∆I/∆B ∝ P0. The quantities are plotted this waybecause they are easily accessible experimentally.
can still measure the photon shot noise level experimentally by independently measuring the total
(summed) power incident upon the photodiodes, P0. Because photo-electrons are produced di-
rectly through absorption of photons, using the wavelength-dependent conversion efficiency of the
photodiodes (similar for all silicone photodiodes), g ≈ 0.6 A/W at 780 nm, the independent current
from each photodiode is Ip = gP0/2, and the noise in the differential photocurrent δIp =√
2Ip
the electrical current will display the same quantized noise,
δIp√∆f
=√
2eδIp (5.19)
= (2)1/4√egP0 (5.20)
The experimentally obtained differential photocurrent per unit field ∆Ip/∆B, either measured
through the normal calibration process or using the central slope of dispersion sweeps like figure
2.12a, can then be used to find the expected photon shot noise
δB√Hz
=
(∆Ip∆B
)−1δIp√Hz
(5.21)
For a typical measured ∆Ip/∆B = 10µA/nT and total probe power of 1 mW, (5.21) evaluates
to δB/√
Hz = 1.5 fT/√
Hz.
92
5.2.4 Averaged parameters
When compared to the derived experimental measurement of the photon shot noise, we have
found evaluating (5.18) with averaged laser parameters from the front of the cell gives unrealisti-
cally low results (compare 0.1 fT/√
Hz to the experimental result, 1.5 fT/√
Hz). The reason is the
spatial dependence of the sensitivity due to the spatial dependence of the pump and probe laser
intensities, both transverse and longitudinal.
The photon shot noise, expressed as δφ, can be calculated knowing just the probe light intensity
incident upon the photodiodes. It would be present without a vapor cell, and is only dependent upon
the cell through the attenuation of the probe light. The conversion to effective magnetic field in
(5.18) utilizes a factor calculated for the magnetometer, the rotation per unit field φ/By, which is
really a function of position.
5.2.5 Spatially dependent magnetometer response
A more detailed method of modeling the magnetometer uses a differential expression for the
probe polarization per unit length and integrates this through the cell, weighting the result by the
amount of probe ligth at each point. A differential and position dependent version of the rotation
per unit field is calculated below. From (2.38) and (5.11)
∂ (φ/By)
∂x=
1
4nref2cD(∆ν2)/B0(x, y, z) (5.22)
=1
4nσ(∆ν2)
(∆ν2
∆Γ/2
)1
B0(x, y, z)(5.23)
=C(∆ν2)
B0(x, y, z)(5.24)
C(∆ν2) =1
4nσ(∆ν2)
(∆ν2
∆Γ/2
). (5.25)
B0(x, y, z) depends upon the complicated propagation effects, and to calculate it requires the so-
lution to the steady state equation (2.23). As discussed in Section 2.4, numerical techniques are
necessary to address (5.23), such as relaxation methods [Wagshul and Chupp, 1994], [Press et al.,
2007, sec. 18.2].
93
The total rotation per unit field is
φ
By
=C(∆ν2)
P2
∫ l/2
−l/2dz
∫ l/2
−l/2dy
∫ l/2
−l/2
I2(x, y, z)
B0(x, y, z)dx, (5.26)
where the probe incident power (P2) and intensity (I2) are normalized such that
1 =1
P2
x
y,z
I2(0, y, z)dydz. (5.27)
The spatial dependence of the probe can be written in advance as (2.49). We make use of the
results developed in section 2.4 to model the pump propagation and Gaussian transverse spatial
dependence. Then, (5.26) can be integrated numerically. Finally, the field term in B0(r) will
depend on the spatial properties of the pump laser through the light shift, as (2.56).
To model the polarization, we use the same technique used in section 2.4. The polarization
is calculated as it would be under diffusion-free conditions, and then an exponential decay to the
walls with characteristic decay length zd from (2.54) is enforced, only this time in all directions.
We make a 3D grid, and setup an initial 2D grid to represent the pump laser intensity profile
incident upon the cell in the x − y plane. Next, each point on this 2D grid is propagated forward
one unit in the z direction, similarly to the 1D case in section 2.4. This process is repeated until
discretized versions of the polarization and pumping rate are obtained for the entire length of the
cell. These functions are interpolated and used to calculate the other parameters of interest. The
fundamental parameter is the figure of merit (FOM), which in our calculations is related to the
integrand of (5.26) by the gyromagnetic ratio scaling,
FOM =I2(r)
γB0(r)(5.28)
= I2(r)Pz(r)Γ′(r)
Γ′(r)2 + Ω(r)2. (5.29)
In our calculations, the decrease in polarization due to diffusion should be accounted for through
the diffusion length, and so the polarization was calculated without any explicit relaxation term
from diffusion, like in section 2.4. However, in calculating the FOM, the correct Γ′ must account
for the total spin-relaxation rate, Γ′ = R + Γpr + Γsd + ΓD, since it describes how quickly a po-
larized atom relaxes. Once the FOM is calculated, the integral in (5.26) is performed numerically.
94
The rotation measurement noise is calculated through (5.14), and combined with φ/By yields the
photon shot noise.
The model parameters used are shown in appendix B. The laser parameters are close to the
optimum used in our experiments. The pump laser is detuned many linewidths from resonance,
which ensures uniform optical pumping but results in a large light shift. Figure 5.3 shows R, ΩLS,
and Pz for the simple model described above. Our diffusion model was designed to be used in high
pressure buffer gas cells [Walker and Happer, 1997], where the diffusion length is very small. It
captures the gross position dependence of the polarization, but none of the polarization smearing
that would occur if a full numerical solution to (2.23) was used. For example, the dimple in Pz
in the center of figure 5.3(b) is due to Γpr, evident by comparing to figure 5.3(d). There is extra
relaxation there, but for our diffusion lengths this would probably be smoothed over.
To illustrate the importance of the light shift for the magnetometer sensitivity, we perform two
sets of simulations, one with this term and the other without. The difference is in the FOM, and
can be seen in figure 5.4(a,c) (ΩLS = 0) and figure 5.4(b,d). The FOM is obviously lower when
including the light shift, leading to a higher photon shot noise (0.8 fT/√
Hz vs 0.2 fT/√
Hz). These
quantities are shown in figure 5.4. The description and simulations presented above are meant to
capture the behavior of the magnetometer, but the roughness of the diffusion model for our low
pressure cells means we expect only order of magnitude agreement between photon shot noise
calculated in this way and the measured value.
Finally, in actual operation, ΩLS is partially cancelled by the application of an opposing real
magnetic field from the procedure in section 4.2. The average gradient ΩLS can be estimated from
figure 5.3(c), and is ±∆ΩLS/w01 on either side of the pump center. We will examine what nulling
field we predict, and how the nulling procedure affects the FOM.
During the nulling procedure, the sensitivity to Ωx is measured and minimized. Considering
the size of the light shift, this sensitivity comes almost entirely from the laser, and the integrated
signal is used. At each position in the cell, we obtain the integrated sensitivity from (2.24) and an
95
(a) R (b) Pz
(c) ΩLS (d) Pz contour plot
Figure 5.3: (a) the pumping rate, (c) the light shift, and (b,d) the polarization as a function of position inthe cell. A simple model is used to account for diffusion and the results integrated numerically. In (d), thedashed red line corresponds to the position of the probe waist. Probe relaxation is seen in the slight Pzdecrease in (b).
96
(a) FOM (ΩLS = 0) (b) ΩLS
(c) FOM (ΩLS = 0) (d) FOM with ΩLS
Figure 5.4: 3D simulations of magnetometer parameters. Comparison of (e,g) and (f,h) shows differencein performance with and without light-shift. The red dashed lines in the density plots show the pump (z)and probe (x) waist profiles.
97
expression similar to (5.26)⟨∂Px∂Ωx
⟩=
⟨Pz(r)
Ωz(r)
Γ′(r)2 + Ωz(r)2
2I2(r)
πw202P2
⟩. (5.30)
Where Ωz(r) = ΩLS(r) − Ω0, and Ω0 is the constant nulling field applied during the nulling
procedure. We adjust Ω0 to minimize (5.30) when using the magnetometer. For our simulations,
after the polarization and pumping rate are found for all points in the cell, we calculate the averaged
response⟨∂Px∂Ωx
⟩→ f(Ω0) and then numerically find Ω0 that minimizes this function. Then the
total field Ωz(r) is used in the FOM integration.
The results for the FOM with the cancellation field applied are shown in figure 5.5. The can-
cellation field was found to be Ω0 = 0.95ΩLS, which means the sensitivity to Ωx is minimized
approximately when then center of the pump shifts the ambient field nearly to zero. The noise
level was 1.1 fT/√
Hz, higher than the estimated value without ΩLS, so the nulling procedure does
not necessarily lead to the lowest noise when the the pump is far detuned.
98
(a) ΩLS-with Ω0 (b) FOM-with Ω0
(c) FOM-with Ω0
Figure 5.5: Similar to figure 5.4, except in addition to ΩLS, we have added Ω0. The result favors atoms inthe center of the pump beam, increasing the overall photon shot noise. (a) shows the total Ωz , and indicatesthe offset is essentially the entire light shifted field. (b,c) show the FOM.
99
5.3 Technical noise sources
As indicated in table 5.1, our sensitivity is limited by fundamental detector noise and real
magnetic fields. This restricts enhancement of MCG SNR through signal processing, because the
techniques usually depend on combining magnetometer channels and reducing correlated signal
corresponding to true but unwanted magnetic field. These techniques fail when the noise is pri-
marily uncorrelated, as in the case of photon shot noise.
Noise reduction methods also benefit from having more channels of information. Parametric
modulation [Li et al., 2006] can provide an extra channel per magnetometer, but at the cost of
reduced sensitivity per channel because of an effective increase in photon shot noise. If the mag-
netometer is dominated by fundamental quantum noise, then this operation mode can decrease
the sensitivity (see section 5.2). However, if the magnetometer is dominated by technical (non-
magnetic non-quantum) noise, this mode of operation can greatly improve the sensitivity, espe-
cially at low frequencies.
The typical noise level of the probe measurement is δIpr ∼10 pA/√
Hz, or at typical I-V gain of
500 nA/V, δVpr ∼5 nV√
Hz. If we are to remain dominated by the actual measurement, electronic
noise must be much smaller than this level.
5.3.1 Signal aliasing
Aliasing is a phenomenon that occurs in any time-dependent system sampled at discrete time
points. If we imagine a system sampled at sampling frequency fs, the maximum frequency that can
be uniquely represented is given by the Nyquist theorem is fmax = fs/2. Signals with frequency
components above fmax will be “aliased” into the baseband frequency range, a processes that maps
any frequency f2 > fs/2 into the set of frequencies 0–fs.
Using our typical fs = 20 kHz, the Nyquist frequency is 10 kHz, and any spectral content
above this frequency can get mapped back into the 0–200 Hz range where we normally examine
the noise floor. To alleviate this problem, we apply filtering on the photodiode amplifiers, but must
100
be careful not to attenuate signals we desire to measure. A compromise of a 2-pole low-pass filter
at 300 Hz is chosen to accomplish these goals.
A second-order low-pass filter above its cutoff point decreases the power by 12-dB/octave.
Since we characterize the noise floor by the amplitude spectral density, the noise contribution to
the noise floor is 6-dB/octave above 300 Hz. This should attenuate any signals above 10 kHz by a
factor of -30 dB (amplitude) or more. The aliased signal is uncorrelated with the baseband signal
to which it gets aliased, and the two powers add in quadrature.
We notice significant electronic noise improvement with the application of these low pass fil-
ters, but some noise is pickup between the photodiode amplifiers and the FPGA system, a line
which is not low-passed. For a quiet environment (no chopped heaters, thermistor measurement
off), using low-noise wiring techniques and ferrite beads on the FPGA inputs wires, the electronic
noise level is 2 pA/√
Hz.
5.3.2 Probe polarization rotation in optical fiber
Polarization maintaining fibers require well-defined linear input polarization along one of two
principle fiber axes. When used properly, the output polarization is stable to the extinction ratio,
typically 20-30 dB. Residual rotation in the fiber leads to a fictitious magnetic field, since magnetic
fields cause the same effect. This can limit SNR at low frequencies where the fiber polarization
changes mostly occur (due to temperature or pressure fluctuations along the length of the fiber).
The difference signal ∆I between a beam of incident intensity I0 at an angle φ − π/4 to the
polarization axis of a polarizing beam splitter is
∆I = −I0 sin(2φ) ≈ 2I0φ. (5.31)
Rotations in the fiber will lead directly to a difference signal which will drift over time. If the ex-
tinction ratio is a full 30 dB, the polarization angle will still drift from the fiber output φ = 1/1000,
we expect the minimum drift of the signal from a 1 mW probe to be δI = 2 µW. A typical magne-
tometer response around ∆I/∆B = 10 pW/fT differential signal would correspond to an error of
about 200 pT from polarization rotation in the PM fiber. This is much larger than the anticipated
101
Figure 5.6: (A) Polarization rotation ∆θ from the PM fiber results in a signal indistinguishable frommagnetic field induced rotations δφ. (B) This effect is largely eliminated by placing an output polarizerafter the fiber, converting polarization fluctuations into intensity modulation.
fMCG signal of interest, with a maximum amplitude of 1 pT and frequency components near DC.
We have taken measures to passively mitigate this problem, such as isolating the temperature vari-
ation in the optical fiber by enclosing it in protective tubing. At the cost of optical simplicity,
another method is placing a polarizer at the output of the probe fiber, a technique utilized in our
magnetometers. The polarization noise from the fiber is converted to intensity noise by the po-
larizer, to which the differential detection is much less sensitive. Specifically, the intensity noise
after the Glan-Taylor is δI = I0δθ2, which gives rise to a measured signal from the differential
photodetector only if there is already a misalignment between the probe polarization axis and the
polarizer φ, for example as might be caused by an actual Faraday signal from a magnetic field. The
difference signal generated (for small angles) is
δI = I0 sin (2φ) δθ2, (5.32)
which is equivalent to a fractional fictitious magnetic field
δB
B= 2I0
(∆B
∆I
∆φ
∆B
)δθ2, (5.33)
where the fractions on the right hand side of (5.33) depend on the calibration of the magnetometer.
With typical ∆φ/∆B = 100 nRad/fT, the worst case scenario for a correctly coupled PM fiber
102
(extinction ratio of 1/100) correspond to ∆B/B = 1/500. The actual magnetic field error depends
on the real magnetic field in the case of a well aligned optical system (δφ = 0 if B = 0), so at zero
field there is no signal change. Finite fields B < 0.5 nT limit the fictitious field to ∆B < 1 pT.
This requirement on the size of the finite field is near the requirement to remain in the sensitive
regime of the magnetometer and does not in reality impose any additional experimental limitation.
5.3.3 Probe frequency and intensity noise
We begin by characterizing the noise from changes in the probe laser intensity and frequency.
Taking the form of the total signal S given in section 2.2.8,
S =P2
2xOD(x)e−OD(x)By
B0
, (2.39)
where x is the usual detuning normalized to the optical HWHM. For what follows, we assume
the probe absorption rate is a negligible component of the total relaxation rate, Γpr << Γ, so the
equilibrium polarization is unaffected by small changes in the probe laser.
Changes in the pump and probe parameters can be expressed by partial derivatives of this
quantity. For example, we may calculate the effective field noise due to intensity fluctuations as
δBy|P2 =∂S∂P2
(SB
)−1
δP2 (5.34)
=δP2
P2
By. (5.35)
This equation assumes the photodiodes are precisely balanced in the presence of no external fields
or pump light. We have not carefully measured the probe laser intensity noise, but we expect it
to be dominated in the frequency region in which we are interested by fiber rotations rather than
actual laser intensity noise, due to the effects described in the previous section.
The probe frequency dependence is
δBy|∆ν2 =∂S∂x
(SBy
)−1
δx (5.36)
=
(1
x− 2
1 + x2+
2x
1 + x2OD(x)
)δx, (5.37)
103
Figure 5.7: Relative frequency noise of the probe for unitBy. At typical probe detuning x 1, the noise isa small fraction of the detected signal for linewidths <10 MHz. Larger very low frequency drift may occur,in which case stabilizing the laser to an external reference may be necessary.
104
which reflects both changes in absorption and rotation. A plot of (5.37) for ∆ν = (1, 10, 100) MHz
and unit By is shown in figure 5.7. For reasonable probe linewidths (< 10 MHz) and normal
operating detuning (x > 10), the relative noise is suppressed by a factor of 100 or more.
5.3.4 Pump laser fluctuations
There is an angle characteristic of the magnetometer, Px = Pzθx, where θx describes the
deviation angle from Pz for an applied field Ωy with the familiar factor
θx =Γ′Ωy
Γ′2 + Ω2. (5.38)
This expression separates the magnitude and tilt contributions to Px. For maximum sensitivity and
assuming no light shift, Γ′ = 2Γ, and θx ≈ 0.2 rad/nT. Then θx . 1 for By . 100 pT, quite a
large field on our scale.
If laser fluctuations are not to add noise to the measurement, the field induced signal must be
larger than the laser fluctuation induced signal. The pump laser only interacts with the measure-
ment through Px, so we just address this term. Using θx, the fluctuations in Px to be compared
are
δPx|Ωy =∂Px∂Ωy
δΩy
= θxPzδΩy
Ωy
δPx|R =∂Px∂R
δR
=
(∂θx∂R
Pz + θx∂Pz∂R
)δR.
In order for pump fluctuations not to limit the magnetometer signal, we want to satisfy the condition
δPx|Ωy δPx|R. After some algebra, this expression is
δΩy
Ωy
R
Γ′
(Ω2 − Γ′(Γ′ + 2xΩ)
Γ′2 + Ω2+
Γ
R
)(δR
R
). (5.39)
The physical parameters that vary are the pump laser intensity and frequency, and (5.39) is a
complicated function of these parameters. For R = Γ and Ω = 0, the sensitivity follows the curve
105
(a) (b)
Figure 5.8: (a) A useful plot for calculating the δR from the physical parameter, δx. The asymptote atR → 2 for x → ∞. (b) δPx due to fluctuations in R. This plot must be bounded between 1 and -1 sincethat covers the range of Px.
in figure 2.9, and it should be evident that at Pz = 1/2, the sensitivity is insensitive to changes in
pumping rate, which agrees with (5.39).
Given the discussion in section 5.2, it is unreasonable to ignore light shifts. The test in (5.39)
requires relating δR to the physical parameters δP1 and δx,
δR2 =
(∂R
∂P1
δP1
)2
+
(∂R
∂xδx
)2
(5.40)(δR
R
)2
=
(δP1
P1
)2
+
(2x2
1 + x2
δx
x
)2
. (5.41)
Hence δR/R is linear in relative power fluctuations and ranges from a factor of 0–2 δx/x. At our
model detuning, x ≈ 20, the relative noise level is δBy/By = δν2 (MHz)/104 from frequency
noise and δP1/P1 for intensity noise.
5.3.5 Laser Misalignment
We examine what happens if the probe laser is no longer orthogonal to the pump and picks up
some sensitivity to Pz. If the probe beam propagates through the atomic vapor at a small angle
θ above the x − y plane, the longitudinal polarization component is now Px + θPz. We expect
θ ∼ 1, which means θ θx in typical low-field conditions. This is a problem if there is extra
noise coupled onto the signal for nonzero θ.
106
Noise analysis proceeds in much the same way as the previous section. Assuming a constant
(non-drifting) angular misalignment, the noise is coupled onto the signal through noise in Pz,
δPx|θ = θ∂Pz∂R
δR (5.42)
= θΓ
Γ′2δR. (5.43)
The field noise can be computed from (5.11),
δBy =θ
θx
Γ
Γ′δR
R(5.44)
= θΓ
γ
(1 +
Ω2
Γ′2
)δR
R. (5.45)
The factor θΓ/γ ≈ 50 pT for our parameters. Disregarding the light shift, this means we need
δR/R 1/(5 104). This is a difficult limit. For example, it means the noise limits due to angular
misalignment coupled to pump noise are
δBy|θ+δν ≈ θδν1 ×(
1 fT
MHz degree
)(5.46)
δBy|θ+δP1 ≈ θδP1
P1
(50 pT
degree
). (5.47)
This calculation ignores the light shift, which for far detuned beams would increase the noise level
significantly for the same misalignment angle. Note that the situation is not the same in nonlinear
magneto-optic effect magnetometers, where the light shift imposes a fundamental limit on the
sensitivity [Fleischhauer et al., 2000, Novikova et al., 2001]. The linewidth of the pump DFB is
expected to be around 5 MHz, but the 1/f fluctuations are likely larger. Moreover, noise from
the pump beam is difficult to diagnose by our characterization methods, since the sensitivity to
background fields is blocked mainly by blocking the pump to test technical noise levels.
To test this, one could measure changes in the signal as the pump frequency or power is dithered
slowly compared to Γ. Then the fluctuations in these parameters could be measured.
107
Chapter 6
Extensions and different operating modes
In this chapter, we discuss extensions to the DC-mode technique pioneered by Allred et al.
[2002] and used in the measurements and discussions presented up until this point. These changes
fall into two general categories: operation modes that do not require the application of large oscil-
lating fields and modes that do.
6.1 Non-parametric field modes
By non-parametric, we mean to indicate methods which do not require the introduction of
modulating fields large compared to the zero field dispersion linewidth in figure 2.12a. We have
used two non-parametric field extensions to the DC-mode operation method. One, which we call
diffusion-mode, uses diffusion to move angular momentum away from the pump beam, where T2 is
larger because there is no pumping light and some of the technical noise problems in the previous
chapter are solved. The second method is feedback mode, in which we use our magnetometer array
as feedback devices with the application of a feedback magnetic field.
108
6.1.1 Diffusion mode pumping
As we discussed in section 3.1 and section 2.2.3, the cells used in this work were originally
filled with an amagat of He, which subsequently diffused through the Pyrex and left us with low
pressure cells. The diffusion length, characterized by (2.54), is around 1.3 cm for our cells (in the
absence of pump light). We expect angular momentum loss to the cell walls to be significant if the
pumping laser is uniformly distributed throughout the cell.
To reiterate the concerns raised in table 3.2, there are three choices to operate a pump with a
waist on the order of the cell size: high pumping rate near resonance (small light shift, high Pz
throughout cell but low sensitivity), pumping rate to get Pz = 1/2 near resonance (small light
shift, but for optically thick cells, the light is quickly attenuated), and pumping off resonance to
get Pz = 1/2 (atoms pumped all through cell, but high light shifts).
The last regime would be the optimum were the pump intensity is uniform, since uniform
light shifts can be cancelled by the application of a real magnetic field. For significant transverse
dependence of the light shift, only the average can be cancelled with a pair of magnetic coils,
and the difference between maximum (center of the pump beam) and the minimum (edge of the
pump beam) approaches the full value of the light shift, which can be quite large and decreases the
sensitivity (discussed in detail in section 5.2.5).
Three approaches can be used to address the problem. First, the pump beam could be made
more uniform, either through expanding its waist or using a spatial filter to make a top-hat beam.
Beam expansion is difficult due to design requirements that favor a compact array. Diffractive beam
shapers may work for this purpose, although they typically create their uniform spatial distribution
only in the working plane of the optic, producing a diverging beam.
Second, the probe beam could be limited so that it only samples the central portion of the pump
laser. However, shrinking the probe beam has the side-effect of decreasing the number of atoms
participating in the measurement, increasing the quantum noise and effective noise from technical
noise sources.
For cells in which the diffusion length can be made large compared to the pump beam, a third
option is to shrink the pump beam and expand the probe beam until the beam waist is approximately
109
equal to the diffusion length, w02 ≈ zD. We call this method diffusion-mode pumping, and discuss
the relevant physics and results obtained with this mode.
We maintain the same power but shrink the pump waist, so the intensity and pumping rate
become large, but only for a small central portion of the cell. In this central region, the polarization
and light shift may be high, but with low sensitivity to magnetic fields. Atoms diffuse from the
central, strongly polarized region to the regions outside the pump beam, remaining polarized until
they undergo a spin-destruction collision.
Using the optics described in appendix B, the pump beam waist was reduced tow01 = 0.51 mm,
and the probe beam waist was increased to w02 = 2.75 mm. The pumping rate becomes huge for
such a small pump beam. At the center of the beam for cylindrical coordinate ρ, R0 increases with
area if the power is maintained constant,
R(ρ) = R0e−2 ρ2
w201 (6.1)
R0 =2Φ01
πw201
σ01(∆ν1). (6.2)
Figure 6.1(top) shows simulations of Pz(ρ) for various R0, but with x modified to make R consis-
tent with our experimental w0 and P1 described by the parameters in appendix B. The normalized
pump profile is also shown. With a narrow waist, the atoms in a region up to the point at which
R(ρ) ≈ Γsd are pumped to high polarization. This occurs at the critical radius,
ρc =w01√
2
(ln
(R0
Γsd
))1/2
.
It can be seen that increasing the pumping rate extends the radius of highly pumped atoms very
slowly. In the limit of P (ρ < ρc) ≈ 1, the pump is linearly rather than exponentially attenuated.
The dots on the normalized pump profile in figure 6.1 show where ρ = ρc for each initial pumping
rate. We calculate Pz by numerical solution to the 2D radial diffusion model in steady-state,
modified from (2.23),
0 = q(P )D1
ρ
∂
∂ρ
(ρ∂P (ρ)
∂ρ
)+R(ρ)(1− P (ρ))− ΓsdP (ρ), (6.3)
and we assume a cylindrical (rather than square) cell, with its axis of symmetry along the pump
the boundary condition P (l/2) = 0. We compare the radial 1/B0 (section 5.2.5) to the theoretical
110
Figure 6.1: (a) Shows the solution to the radial diffusion problem for various starting pumping rates. (b) Aplot of B−1(ρ)/B−1
0,opt., comparing the sensitivity per unit radius at each point in the cell compared with auniformly pumped cell when R = Γ.
111
optimum if Pz = 1/2 everywhere. While the numerical solution gives the stead-state Pz, we must
still account for diffusion losses in the relaxation rate of Pz when calculating the radial 1/B0,
and use the effective decay rate ΓD introduced in section 2.2.3. Figure 6.1(bottom) shows the
normalized B0,opt./B0, neglecting light shifts (these are only in the central portion of the cell,
where 1/B0 is already small). At ρ > ρc, the polarization is large but the pumping rate is low, so
1/B0 is actually higher than the uniform pumped case for some cell positions. What will matter
for noise purposes, however, is the FOM from section 5.2.5.
The previous simulations of pump propagation, shown in figure 6.1, are informative here.
While the attenuation will be greater at the front and rear cell walls (all the atoms are sinking
angular momentum, only a few see a high pumping rate), the high pumping rates ensure relatively
uniform polarization through the cell and shorten the effective diffusion length inside the pump
light cylinder. Very high pumping rates allow us to ignore diffusion in the z-direction and the po-
larization will be uniform along the z-direction. For this simple model, we set Pz(ρ > l/2) = 0,
and use the polarization profile P (r) = Pz(√x2 + y2) for all z values in the cell. The FOM is
found, and the results for different R0 are displayed in figure 6.2.
The integrated FOM comes out significantly higher than the uniform case for Pz → 1/2. At
our operating parameters,y FOM
FOM|Pz→1/2
dV ≈ 11
and the expected photon shot noise, calculated as in section 5.2.5, is δBpsn = 70 aT/√
Hz. The
main advantage is apparent when comparing figure 6.2 to figure 5.5. Confining the pump light
without confining the polarization moderates the light shift and pump absorption issues that make
achieving R = Γ difficult in the entire cell volume. The portion of the cell with good sensitivity to
magnetic fields has increased.
The noise estimate is probably unrealistically low, and certainly lower than we have actually
achieved. The spin-decay rate ΓD we have used is known to be the exponential contribution to
spin-relaxation with both exponential and non-exponential terms, and the effective relaxation rate
should be higher Wagshul and Chupp [1994].
112
Figure 6.2: Models of diffusion mode pumping FOM for a variety of initial pumping rates. For the data inthis thesis, the pumping rate was closest to R = 104×Γsd. The green circles have radius ρc, the red-dashedcircles in the x-y plane have radius w01 and in the y-z plane have radius w02.
113
A more careful analysis could be made by solving (2.23) using a diffusion mode expansion. ΓD
used in this analysis corresponds to the fundamental diffusion mode, but higher modes have faster
decay rates. The solution to Pzi can be characterized for each mode (i), with its own relaxation
rate ΓDi , and the contribution to the sensitivity from each mode summed,∑
i Pzi/ΓDi . Such an
analysis is being investigated at the time of this work.
Diffusion mode pumping requires zd =√qD/Γsd > w01. For cells of our size, this leads to
relatively large wall relaxation ΓD. It may be that high pressure cells compensate the difficulty of
pumping a cell uniformly without large light shifts by their lower Γ.
In our experiment, diffusion mode pumping did seem to outperform a larger pump beam. Our
best results were obtained using this method, and are shown in figure 6.3. Low frequency contri-
butions to the noise profile are especially suppressed, and the usual 6.7 Hz peak can be seen with
its harmonics.
Finally, because the pump light is hardly attenuated through the cell, there is significant pump
power that must be dumped somewhere at the back of the cell. We have lined up two magnetome-
ters and pumped them with a single beam, with similar noise performance, and it appeared we
could have pumped up to four magnetometers with one beam. This might be a significant advan-
tage in constructing a large magnetometer array, reducing significantly the optics necessary for the
pump beam.
6.1.2 Feedback mode
Magnetic field feedback is a natural extension to the DC-mode magnetometer. For one thing,
the DC-mode magnetometer signal can be used directly (by negation) as an error signal for By.
Feedback is as easy as plugging the output of the I-V converter into a set of feedback coils, at least
for that direction.
Magnetic field feedback has been used in previous experiments. The originalMx-mode magne-
tometers used feedback [Bell and Bloom, 1957] to measure the magnetic field. Bison et al. [2009]
used a complicated feedback scheme to operate a 25-channel array of atomic magnetometers in
114
Figure 6.3: Noise characterization for the magnetometers operating in the diffusion pumped mode. Themagnetic noise (black), probe noise (blue), and electronic noise (red) are overlaid to give a better indicationof which are dominant. The probe noise level on channel 3 is deceptive, since it remains sensitive to fieldseven with the pump blocked.
115
the Mx mode and make them an effective 19-channel gradiometric array. Their objective was sup-
pression of ambient noise in an eddy-current shield, and their second order gradiometer achieved
a suppression ratio of 1:1000. Seltzer and Romalis [2004] built an unshielded magnetometer that
operated in the SERF regime through 3-axis feedback, using a single DC-mode magnetometer to
measure all three fields simultaneously. The process is equivalent to the one we use to null the
fields in the first place (section 4.2). The signal is proportional to Ωy, and if small modulating
fields are added Ω(x) = Ω1 sin(ωxt) and Ω(z) = Ω1 sin(ωzt), then using lockin detection on ωx
(ωz) measuresBz (Bx). These were all fedback to large Helmholtz coils surrounding the cells used
in the experiment, and an unshielded noise level of around 1 pT/√
Hz in the difference signal was
obtained. Similarly to Bison et al. [2009], Seltzer and Romalis [2004] described and implemented
feedback primarily as a background field noise-reduction technique, and to our knowledge, SERF
feedback has not been investigated in a MSR-type environment. However, it has many potential
benefits for biomagnetism applications.
First, many of the technical noise sources described in section 5.3 are proportional to the overall
signal. As one example, the noise from intensity fluctuations is proportional to the difference signal
of the photodiodes. Feedback can be used to maintain zero difference signal in the presence of
fluctuating magnetic fields, suppressing this noise source.
Second, the primary goal of fMCG is the accurate measurement of B(t), which allows the
extraction of medically relevant information. Since the fMCG signal has frequency elements that
extend through the changing magnetometer frequency response, fMCG measurements (and any
measurement where accurate B(t) reconstruction is important) must be deconvolved with this
response. Figure 4.1 shows a typical example of such a calibration and the fits used to characterize
the response, and clearly these fits are not accurate to 1%. Averaging more response measurements
would help if the response was stationary, but offset fields change frequently enough that this is
impractical.
Furthermore, many signal processing techniques used in fMCG rely on measurements across
multiple magnetometer channels. For example, simple gradiometry in the DC-mode setup would
mean measuring an fMCG signal at multiple channels, deconvolving each measurement with the
116
Figure 6.4: Block diagram of the feedback loop used to cancel By for each magnetometer. When the largefeedback coil is used, the control box is replaced by a resistor, but otherwise the loop is the same.
response from each channel, and subtracting the result. For perfect measurements, a uniform
field at two channels would give zero difference after this procedure. With a 10% error in the
calibration, the suppression factor will depend on frequency and we wouldn’t expect better than
∼ 85% suppression at any frequency. More advanced techniques like PCA would work well
on unknown calibrations but primarily if they are spectrally flat. In feedback mode, the signal
measured becomes the current applied through the feedback coils to maintain zero field at the
sensor. The feedback field is a linear function of current, so measuring the current gives a flat
frequency response, and only an absolute calibration to relate the measurement to the ambient
field.
Finally, if we need to detect anything in real time, the background magnetic fields in the MSR
are quite large and can make this difficult. The 6.7 Hz peak alone has an amplitude between 5 −
20 pT, dwarfing the smaller fMCG features, and the mother’s MCG will have a similar amplitude
at the array position. Both sources are far from the array relative to the fetus, and so their fields
are comparatively uniform across all channels. This calls for real-time gradiometry, but that is
complicated by the requirement of real-time deconvolution.
We have used two methods of feedback. In the individual feedback mode, the magnetometer
signal is conditioned through a PID loop and sent to the coil control box for each magnetometer,
which uses the individual calibration coils for each magnetometer to maintain zero field. Figure
6.4 shows a block diagram of this technique. The output voltage Vo is a conditioned measurement
control voltage for the voltage controlled current source output. As self-zeroing magnetometers,
117
the feedback lowers technical noise and flattens the frequency response, making real time mea-
surement easier. However, the current electronics seem to increase the noise level significantly,
and we have yet to obtain good results with this method.
A more useful improvement is using a single large feedback coil, as in figure 6.5, which we
refer to as gradiometer feedback. The magnetometers are placed symmetrically about the center
of the coil so that the magnitude of a field applied by these large coils is the same. One reference
magnetometer, channel i, provides the error signal to this coil. Assuming the nulling process
initially sets the field at each magnetometer to zero, feedback maintains the field at each channel
Bj(t) =
0 j = i
B0j(t)−Bi(t) j 6= i
for local ambient field B0j and total field with feedbackBj . Typically there are different offsets for
each channel applied locally to the calibration coils, adding a different DC term for each channel.
The great benefit is time-changing uniform magnetic fields get cancelled at each magnetometer,
enhancing the SNR from nonuniform fields between the reference and other channels. Examining
the noise budget table 5.1, the dominating magnetic noise is from circuit noise in the nulling coils,
which will be uncorrelated, as is most technical and all quantum noise. When utilizing feedback
mode, we expect the noise floor to modestly increase due to this uncorrelated noise, but to get a
large decrease in correlated magnetic noise in the non-reference channels. This behavior is seen in
figure 6.6, where the low frequency noise is mostly well cancelled, especially the uniform 6.7 Hz.
The advantages in real-time measurement are most easily seen when there is a signal of interest
that we wish to isolate from background fluctuations. Fetal MCG was taken with this technique,
and we present a portion of it in figure 6.7. This data corresponds to a 5 min continuous fMCG
measurement, taken using the gradiometer feedback method with channel 4 (bottom green) as the
feedback reference. The objective for real-time detection is signal optimization, and so we apply
a 1-30 Hz 2-pole bandpass filter to help isolate the fMCG QRS signal from background. This
filter was applied to the data during post processing, but the same filter can be applied in real-time
using the I-V converter front-panel controls. During the initial part of the measurement (left), the
118
Figure 6.5: The magnetometer array, with the single rectangular coil. The magnetometers are placed sym-metrically around the center of the coils, and the coil symmetry ensures the field seen by each magnetometeris the same. One magnetometer, i, provides the error signal, and feedback maintains Bi = const. The fieldsseen at the other magnetometers is Bj 6=i = ci + (B −Bi), where ci is the constant offset applied locally toeach magnetometer through its cancellation coils. If the fields are initially nulled, ci = 0.
119
Figure 6.6: A plot of the ratio of spectral noise amplitudes between feedback off and feedback on. Numberson this scale below 1 correspond to decreased noise, numbers above zero are increased noise. Channel 1is not a field measurement, but rather monitoring the error signal, and is the performance of the feedbackscheme.
120
HVAC fan was turned off, and the fMCG QRS complex is easily identified in all four channels.
The mother’s MCG signal is better cancelled in channel 3 than in channels 1 and 2. This is a result
of the orientation of the array during the measurement, where channels 1,2 were closest to the
mother’s heart and 3,4 farthest (see figure 4.5).
Later in the measurement (right), the fan is turned on, and channel-4 responds to keep the field
zeroed. In the feedback signal, the originally clear mother’s QRS complex is barely discernible,
and there is no sign of the fMCG QRS complex. In the other three channels, the fetal QRS complex
is still seen.
Operation of the feedback loop is relatively straightforward, since the error signal is exactly the
magnetometer signal. We follow the following procedure for using a magnetometer in feedback:
• Set up magnetometers with nulled DC fields
• Connect the desired reference magnetometer to the PID circuitry and connect the PID cir-cuitry to the feedback coils
Large coils: The feedback signal is applied across the coils and a suitable series resistor.
Individual coils: Feedback voltage is applied to the input of the coil control box.
• The polarity of the feedback path can be adjusted by the “invert” output of the I-V converter–if the magnetometer output drops to zero or oscillates, the output is connected with thecorrect polarity.
• Set the I-V converter gain and frequency cutoff, using only a one-pole filter for the cutoff.Decrease the P-gain until the system no longer oscillates.
• With P-gain set, increase I-gain until the system oscillates, then decrease it back below thispoint. We generally do not use D gain.
Finally, we provide a general analysis of the feedback elements to understand the limits of the
system, how they linearize the magnetometer response, and what issues we will have to overcome
to generalize feedback in z-mode. The magnetic field will be suppressed by the closed loop gain
as
btot(ω) =ba(ω)
1 +Gcl(ω), (6.4)
whereGcl = Gmag(ω)GI−V(ω)GPI(ω)GboxAcoils is the closed loop transfer function. WhenGcl(ω)
1, the feedback field is equal to the ambient field, with any frequency dependence suppressed. For
121
Figure 6.7: Data is from a continuous 5 min fMCG dataset in which the HVAC fan was turned on duringthe measurement. The bottom channel is the gradiometer feedback reference. The other three channels arelargely unaffected by the fan turn on, and both the mother’s and fetal MCG QRS complexes are clearlyvisible in real time both before and after the fan is turned on. The signals have been processed with abandpass filter from 1-30 Hz, easily implemented in real time using the I-V front panel controls.
122
the large coils, Ac = 3×109 fT/A and is used with a resistor, Gbox = R. A typical calibration from
the magnetometer is Gmag ≈ (5 pA/fT)fm(ω), where fm(w) is the normalized frequency response
of the magnetometer, characterized in figure 4.1. Both GPID(ω) and GI−V(ω) are adjustable. For
noise reasons, we prefer maximizing the amplitude of GI−V(ω) → GI−Vfiv(ω), and adjusting
fiv(ω) to maintain loop stability using a single pole low-pass filter with controllable settings via
the I-V converter front-panel.
Near DC, the suppression of the large fan noise is important. To reduce the fan noise to
∼1 fT/√
Hz, we need Gol(6 Hz) ≈ 105. As with any negative feedback loop, the gain must be
rolled off so that Gωc < 1 at the frequency ωc where the feedback loop components induce a
180 phase shift [Moore, J. H. and Davies, C. C. and Coplan, 2003, p 481]. In DC-mode, fm(ω)
has a 3 dB frequency around 50 Hz. If fm(ω) is constant until high frequencies, the magnetometer
roll-off is usually enough to suppress oscillations until the gain is turned up so much that the funda-
mental I-V converter bandwidth is important. However, if we want to use z-mode, we must ensure
the feedback does not cancel the response from the z-mode modulation frequency, or else the sen-
sitivity to the fundamental component of the signal will be suppressed. If the frequency must be
rolled off by a linear filter fm(ω), the maximum roll off form the combined fm(ω)fiv(ω) filter is
-20 dB/decade (amplitude). This would make the maximum gain at 10 Hz Gol(10 Hz) = 104, and
would practically need to be less to maintain nonzero phase margin and to perturb the response
at 1 kHz less. Lower ambient field suppression or nonlinear filtering might still allow z-mode
feedback.
Measurement by the reference channel in feedback mode is made by monitoring the output
voltage from the PID circuit, VPID. Because flexibility is lost in the choice of antialiasing filters,
the detection circuit has a low-pass antialiasing filter at 1 kHz, as well as 10x gain to minimize
electronic measurement noise.
6.2 Parametric field modes
This entire thesis, to this point, has focused on the DC operation of the magnetometer, which is
characterized by the steady-state response (2.24) and low frequency perturbations from oscillatory
123
fields, where Px follows Ωy(ω), but with diminished amplitude and larger phase lag at higher
frequencies.
Modulation methods utilize relatively large oscillating magnetic fields to modulate the atomic
polarization, where these fields are no longer a perturbation on (2.24), and in fact dominate the
response, which is best characterized in terms of the modulation frequency. We discuss two such
modes in this work: z-mode and transverse pumping mode.
Z-mode pays a small price in sensitivity for simultaneous measurement of two field directions
and higher frequency modulation [Li et al., 2006]. If the magnetometer is dominated by technical
(non-magnetic and non-quantum) noise, this mode of operation can greatly improve the sensitivity,
especially at low frequencies. Each measured field direction is counted as a measurement chan-
nel for signal processing purposes, and doubling the number of channels without increasing the
number of cells is a potentially important boon for multichannel signal processing techniques.
A second modulation mode utilizes pumping transverse to a large magnetic field. Large am-
plitude but short time period 2π pulses are added along the transverse field direction, imposing a
resonance on the transverse spin when the 2π pulse cycle frequency matches the transverse field.
This magnetometer, which we call a transverse pumping magnetometer, can still operate in the
SERF regime. Combined pump/pulse modulation can be used to control the precession frequency
of the atoms even in the presence of large DC fields.
6.3 Z-Mode
In z-mode, the additional parametric field is a large oscillating field along the pump direction,
Ω = Ωxx + Ωyy + (Ω1 + Ω0 cos(ωzt)z) and the applied field Ω0 is large, Ω0 Ωx,Ωy. The
details of this technique were first presented in Li [2006], Li et al. [2006], but we present a similar
derivation here because the methods used in the analysis of the transverse pumping magnetometer
are identical.
We assume spin temperature still holds and neglect hyperfine structure and diffusion, and as-
sume the same magnetometer geometry as in figure 2.6. We will also ignore the dependence of the
slowing down factor on P . The steady state equation (2.23) can be rewritten in terms of rotating
124
polarization (P± = Px ± iPy) and fields (Ω± = Ωx ± iΩy), e.g.
qP+ = −Γ′P+ − iΩ+Pz + iΩz(t)P+ (6.5)
qPz =1
2i(P+Ω−P−Ω+)− Γ′Pz +R (6.6)
with Ωz(t) = Ωz + Ω0 cos 2ωzt. If the transverse fields are small, then Pz = R/Γ is a steady state
solution, as before. Equation (6.5) can be solved with the ansatz P+ = A+eiφ(t). Plugging this into
(6.5) gives
A+ = −(
Γ′
q+ i
(φ− Ωz(t)
q
))A+ − i
Ω+Pzq
e−iφ(t) (6.7)
which suggest the choice φ = (Ωz(t)+ iΓ′)/q, which can then be integrated using the explicit form
of Ωz(t) to get
φ =1
q
((Ωz + iΓ′)t+
Ω0
ωzsinωzt
). (6.8)
Assuming Ω+ φ, it is approximately constant and (6.7) can be directly integrated,
A+ = −iΩ+Pzq
∫dt′e−iφ(t′) (6.9)
= −iΩ+Pzq
∫dt′e(Γ′−iΩz)t′/q−iu sinωzt′ (6.10)
= iΩ+Pz
∞∑m=−∞
Γ′ + i(Ωz +mqωzΓ′2 +mqωz
Jm(u)e(Γ′−iΩz)t/q−imωzt (6.11)
where u = Ω0/(qωz) and the last line makes use of the Jacobi-Anger expansion,
eiu sinωzt =∞∑
m=−∞
Jm(u)eimωzt. (6.12)
The expression for P+ is given by our original ansatz,
P+ = iΩ+PzΓ′
∑m,m′
FmJm(u)Jm′(u)e−i(m−m′)ωzt (6.13)
Fm =1 + i(Ωz +mqωz)/Γ
′
1 + (Ωz +mqωz)2/Γ′2. (6.14)
If Ωz = 0, the term Fm has a 2nd order bandpass filter response with cutoff frequency |m|ωz =
|Γ′ − Ωz|/q, which is centered at m = 0 and rolls off the importance of higher |m| terms.
125
It is enlightening to Fourier expand the oscillator components of (6.13). The DC term is found
by requiring m′ = m in the sum of (6.13), and terms oscillating at nωz are found by adding
δm,n±m′ei±nωzt to the sum. After expanding the complex exponentials, the response is
P+(DC) = iΩ+PzΓ′
∑m
FmJ2m(u) (6.15)
P+(nωz) = Ω+PzΓ′
∑m
FmJm((Jm−n − Jm+n) sinωzt+ i(Jm+n + Jm−n) cosωzt) (6.16)
(the argument u for the Bessel functions has been suppressed for compactness, Ji ≡ Ji(u)). Fi-
nally, our probe beam is sensitive to Px = Re(P+). Qualitatively, the DC response will be sensitive
to Ωy, unless a static Ωz mixes in sensitivity to Ωx, the same behavior in DC-mode. For the AC
cases, (6.16) indicates
Px(nωz) =∑m
PzΓ′2 + (Ωz +mqωz)2
Jm×
[Ωx (Γ′(Jm−n − Jm+n) sin nωzt− (Ωz +mqωz)(Jm+n + Jm−n) cos nωzt) +
Ωy (−(Ωz +mqωz)(Jm−n − Jm+n) sin nωzt+ Γ′(Jm+n + Jm−n) cos nωzt)] .
(6.17)
Limits to these equations were previously discussed when the modulation frequency was large,
ωz Γ′ and the DC fields small (Ωz Γ′) [Li et al., 2006]. Then only the m = 0 term is
important because of the fast fall off of Am for higher m. Making use of the identity J−m(u) =
(−1)mJm,
Px(DC) ≈ωzΓ′
−PzΓ′
ΩyJ0(u)2 (6.18)
Px(nωz) ≈ωzΓ′
−PzΓ′
2J0(u)Jn(u)
−Ωx sinnωzt n ∈ odd
Ωy cosnωzt n ∈ even(6.19)
Experimentally, the response of Px(DC) can be obtained by low-pass filtering the modulated sig-
nal, while the Fourier components nωz can be detected by standard demodulation methods (e.g.
using a lock-in amplifier). Under the high frequency, low DC field assumption for (6.18) and
(6.19), then the DC and n = 1 terms can be used to simultaneously and independently measure
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Figure 6.8: Amplitude of the first three Fourier terms which modify the DC sensitivity when ωz Γ′.When using the DC and ωz terms to detect Ωy and Ωx, the optimum is where two Bessel products cross, atu = 0.9. The sensitivity for each channel is 80% of the DC-mode sensitivity.
two components of the magnetic field, with a sensitivity modified from the original DC-mode by
the Bessel function pre-factors J0(u)2 for Ωy and 2J0(u)J1(u) for Ωx, which can be seen by com-
paring (6.18) and (6.19) with (2.24). As discussed in Li et al. [2006], the n = 2 term could also be
used for Ωy detection, allowing lock-in detection of both signals, but J0(u)J1(u) and J0(u)J2(u)
have a much lower common optimum. Figure 6.8 shows these first three terms, and the best dual-
channel operation is where the DC and n = 1 Bessel product terms cross. The sensitivity here is
still 80% of the original DC-sensitivity.
One complication is that, with our high Γ, Γ′ ≈ 103 s−1. Since the optimum response is where
u ∼ 1, the amplitude of the modulating field should be Ω0 ∼ qωz, however Ω0 should be kept low
enough to maintain SERF operation. This limits ωz 2π 10 kHz for T = 145 C. We typically
operate at 1 kHz to compromise between the desire for higher frequency and increased relaxation
rates due to higher SE relaxation. These frequencies are sufficient so that the Fi≥1 F0.
Experimentally, Z-mode requires a slightly different nulling procedure. We begin with each
magnetometer zeroed by the DC nulling process described in section 4.2. Imperfect initial photo-
diode balance and elliptical probe polarization can affect the DC offset, which does not affect our
DC-mode operation but is important to ensure response at each frequency component aligns with
the coil axes. For example, when the initially set DC values are used in z-mode, the response to
the fundamental frequency ωz typically includes By at the 10% level of Bx. While the different
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frequency components are still orthogonal to one another, this indicates bias fields are still present,
and the procedure below can be used to cancel them.
Ensuring the anti-aliasing filters have been increased to 3 kHz cutoff frequency, the z-mode
field is applied at the same frequency ωz ≈ 2π × 1 kHz to each magnetometer individually. Cur-
rently, we use a set of FPGA outputs connected to the z-axis coil control box input, and change
the control box gain to obtain suitable signal sizes. Two small test fields are used alternately,
Ωx = a sin(ω1t) and Ωy = a sin(ω1t), of similar size to the calibration fields used in previous
sections, and adjustments to the DC offset fields B0x, B0y, B0z are made as follows:
• Assuming we will use the low-passed signal to measure By, adjust Ω0 to maximize the
response to Ωx.
• Adjust B0x to eliminate any sin2(ω1) behavior in in the fundamental response. Adjust B0z
to minimize the low-pass response while maximizing the fundamental response.
• Now watching the fundamental response, adjust B0y to eliminate any sin2(ω1) behavior.
Readjust B0z to minimize the low-pass response while maximizing the fundamental re-
sponse.
• Readjust Ω0 to ensure the fundamental response to Bx is maximized.
• If large changes in the offset fields were made during any step, this process should be re-
peated.
After the nulling procedure described above, the low-pass and fundamental response are sensitive
to the “wrong” field to < 5%. The two different magnetometer axes each require independent
calibration. Calibration steps are the same as described in section 4.3.2, but occur in two separate
steps, for each direction.
Figure 6.9 shows z-mode noise spectra. The probe noise (not shown) dominates the noise
floor in these modes, mostly because the I-V converter gain must be turned down or else the large
modulation field saturates them, and because the anti-aliasing filter frequency must be increased.
Still, the noise floor from all channels is <20 fT/√
Hz.
Expanding feedback to this mode requires some care, since the independent field components
are separated in frequency space. Separation of the two signals Bx and By by electronic filtering
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Figure 6.9: Noise spectra obtained using z-mode, using the DC and n = 1 components to measure Byand Bx. The red and black dashed channels correspond to measurements of Bx and By respectively. Thehorizontal blue lines are references to guide the eye. Their values, reading from top to bottom (channel 1 tochannel 4), are 10–20 fT/
√Hz.
129
would require a filter with a cutoff steep enough to isolate 100 Hz from 900 Hz, and with a gentle-
enough phase response that the loop was stable, and with gain high enough to usefully suppress the
background fields. This is a challenging filter, and we have not found a suitable implementation
of it. Using the demodulated or low-passed signal introduces phase lags in the feedback path,
requiring reduction of the signal gain in order to maintain stability. While a solution might exist that
allows feedback to maintain both Bx and By at zero, a workaround is to use three magnetometers
in z-mode, and use the fourth as an error signal to the big coils in feedback mode. This compromise
allows allows the benefits from feedback in By while only decreasing the channel count from 8 to
7.
6.4 Transverse pumping SERF magnetometer
This section outlines a new method of SERF magnetometry, which takes advantage of spin-
resonance in transverse pumping for high magnetic fields. While we describe the transverse pump-
ing magnetometer in the context of SERF magnetometry, the basic technique is useful in any sit-
uation where one wants to polarize an alkali vapor perpendicular to a very large magnetic field,
and can be extended to situations where one wants to control the alkali precession frequency, say
to match the precession frequency of another precessing atom intermixed with the alkali. Previous
studies have investigated this phenomenon at low transverse field amplitudes [Cohen-Tannoudji
et al., 1969, Slocum and Marton, 1973]. To our knowledge, none have investigated it in the context
of SERF magnetometry.
Before describing the transverse pumping magnetometer, we point out two limitations to any of
the SERF techniques described above that this new method overcomes. First, practical vapor cell
issues limit the maximum attainable Rb density to n < 1015 cm−3 at 200 C. As we discussed in
section 2.1.2, this limitation imposes an upper bound in the ambient field magnitude in which SE
relaxation is suppressed. For a 87Rb cell at 200 C, the SERF condition is |B| 4µT, which would
prohibit SERF magnetometry in Earth’s field environments. Feedback could be used to lower the
ambient field into the SERF regime [Seltzer and Romalis, 2004], but this would be unsuitable
for situations where the DC field is required, such as NMR applications where the DC field helps
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increase spin-coherence times of the NMR sample. The transverse pumping magnetometer enables
SERF operation in the presence of high DC fields. In particular, SERF operation in Earth-sized
fields is feasible, and would open the possibility of extending SERF magnetometers to a host of
new applications.
A second practical limitation is the need for calibration. Section 4.3.2 describes the calibra-
tion procedure, which enables a field measurement to be obtained from continuously monitoring
alkali spins. This procedure is, for the most part, sufficient when absolute accuracy is unimportant
(though see caveats in section 4.3.2 and section 7.2). This is the case in most biomagnetism appli-
cations, where the desired information is in the timing of signals rather than their overall amplitude.
Magnetometers that detect an actual atomic precession frequency, such as those utilizing the Mx-
technique, measure a frequency related through fundamental constants to the magnetic field, no
external calibration necessary. Again, for SERFs, feedback can help (see section 6.1.2), but still
requires accurate calibration of a set of magnetic coils. The transversely pumped SERF magne-
tometer can be used in closed loop mode as a resonator, with its resonance frequency determined
by a Larmor frequency.
The geometry is in figure 6.10(a), where the pump and probe beams are still orthogonal, but
there is a very large field orthogonal to both. Since this field is the largest interaction with the
atoms, it defines the z axis, and for this section we use a different set of coordinates, marked in
figure 6.10(a).
The experimental scheme is now the fields in the transverse direction should be zero, but there
is an applied large Ωz and a periodic Ω1(t) with period ω1, Ω = (Ωz + Ω1(t))z. The pump is in the
x-direction. The Bloch equations, assuming spin-temperature and our new geometry, are the same
as (6.5), except with the addition of a pumping term. For a CW pump laser,
qP+ = −i(Ωz + Ω1(t))P+ − Γ′P+ − iΩ+Pz +R (6.20)
where P+ and Ω+ have the same definition as in the previous section. If the transverse fields are
zero, Ω+ = 0 and Pz = 0. The choice of the form of Ω1(t) is important. The optimum, for
the case where we want to operate the magnetometer in the SERF regime, will not be sinusoidal
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Figure 6.10: Diagram of the transverse pumping setup described in this section. Transverse pumping allowsPx ≈ 1, despite R Ωz .
modulation. However, to start with we will use Ω1(t) = Ω1 cosω1t. Then the equation of motion
for P+ is
qP+ = −i(Ωz + Ω1 cosω1t)P+ − Γ′P+ +R. (6.21)
This equation can be solved in exactly the same manner as (6.5), with the substitution from that
solution −iΩ+Pz → R,
P+ =R
Γ′
∑m,m′
FmJm(u)Jm′(u)e−i(m−m′)ω1t (6.22)
Fm =1 + i(Ωz +mqω1)/Γ′
1 + (Ωz +mqω1)2/Γ′2. (6.23)
with u = Ω0/(qω1). The DC part for both spin components is once again found when m′ = m,
Px =R
Γ′
∞∑m=−∞
J2m(u)
1 + (Ωz +mqω1)2/Γ′2=R
Γ′
∑m
J2m(u)
1 + ∆Ω2m/Γ
′2 (6.24)
Py =R
Γ′
∞∑m=−∞
J2m(u)(Ωz +mqω1)/Γ′
1 + (Ωz +mqω1)2/Γ′2=R
Γ′
∑m
J2m(u)∆Ωm/Γ
′
1 + ∆Ω2m/Γ
′2 , (6.25)
where in the right hand side we have made the substitution ∆Ωm = Ωz +mqωz, intended to make
the resonance nature of the equation clear. Most of the physics of the transverse oscillator is in the
above two equations, and they are plotted in figure 6.11 for Γ′ = 0.1Ωz First, if we imagine using
the DC term as a magnetometer, the only field in the problem is Ωz. If we look at the behavior of
Px, we see that there is a resonance for every m value, ω1 = Ωz/(mq). At this resonance, Px is
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(a) DC Px response (b) DC Py response
Figure 6.11: (a) The response along the pump beam, for good pumping R ≈ Γ′ and two different fields,Ωz = 10, 100Γ′. (b) Py for the same situations.
maximized for a single Jm(u) term. The resonance at zero frequency is still the largest, but others
show up at integer m. When not on resonance, the small pumping rate cannot pump the atoms.
When on resonance, the motion of the atoms is complicated, but the sum of the modulation and
DC fields conspires to allow the atoms to precess more slowly when oriented along x than −x.
Py has a similar dispersive-like curve as the DC case, but again modified to have many reso-
nances rather than a single one at zero frequency. The sharp central feature is a function of the
relationship between Ωz and ω1, and if we use a constant modulating frequency, small changes in
Ωz will be proportional to the difference from resonance. In fact, each term in the sum (6.25) is
exactly the same equation as (2.24) if Ωy → ∆m, except for the J2m(u), which just modify the
amplitude. All of the results developed in chapter 2 can be adapted to the behavior here in the
transverse pumping case.
The structure of the modulating field becomes important here. If we wish to maximize the time
spent aligned along the pump axis, then a repetitive pulse with the shape in figure 6.12a(b) is the
best to use. The shape is approximately a periodic 2π pulse, with repetition frequency ω1 = Ωz/q,
and here q = (2I + 1), where I is the nuclear spin. We will discuss why q is independent of the
polarization shortly. It is important to remember though that the actual precession rate of polarized
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(a) Optimum Ω1(t) (b) Motion of polarization
Figure 6.12: (a) Optimum shape for Ω1(t). The AC-coupled pulse (in red) consists of two parts. Initially,the amplitude of the pulse is ΩL for most of the periodic waveform. Then a large amplitude Ω0 is appliedfor a time Ω0d T = 2π. When the pulse frequency is 2πΩz/q, then ΩL = −Ωz , and these fields cancelout during the low portion of the pulse, and the SERF regime is maintained. For very short pulses, spinrelaxation is suppressed because d T Tse (few collisions occur during the high field). The SERF regime ismaintained during the entire pulse, despite the field averaged over a single pulse remaining large (Ωz R).(b) Motion of the polarization components, under the assumptions described above. The shorter d, thehigher the maximum Px.
134
atoms for a physical field B is Ω = γB/q. The pulse is AC-coupled, thus over a single cycle the
pulse area is enforced to be zero, and the averaged DC field is Ωz, as it is without any pulse. This
implies the motion of Rb atoms over one pulse cycle is always TΩz/q.
In terms of the low (high) amplitudes ΩL(H) and the low (high) time periods TL(H), and the total
period T = TL +TH = 2π/ω1, there are three governing relations for the waveform in figure 6.12a:
ΩL
qTL +
ΩH
qTH = 0 (AC coupled) (6.26)
dΩ1
qT = 2πx (Pulse area 2πx) (6.27)
T =2πq
Ωz
(Resonant pulse frequency). (6.28)
These equations allow a solution for the low field under the conditions prescribed above (AC
coupling and at resonance frequency),
ΩL = − x
1− 2dΩz. (6.29)
For the limiting case of an infinitely short pulse with 2π area, ΩL = −Ωz.
This is a remarkably useful result: by making a short AC coupled 2π, and repeating that pulse
at the resonance frequency, the total field TL (Ωz+ΩL) is zero. For an infinitely short pulse, Ω = 0,
except during the pulse itself. The best pulse shape will delay the precession during the cycle until
the pulse at the very end, during which the precession occurs rapidly. When d 1 but finite, there
is a pulse area 2π(1− 2d) for which ΩL = 0 still. Hence, SE relaxation can always be suppressed
during the long part of the cycle.
During the high part of the pulse, SE collisions can still lead to relaxation. This problem is
mitigated by two effects. First, if the atoms are highly polarized, SE relaxation is suppressed since
the atoms are pumped into the stretched hyperfine state [Savukov and Romalis, 2005]. Secondly,
on resonance, T = q 2π/Ωz, so the “on-time” for the high field is TH = 2πqd/Ωz. If the pulse
width is much shorter than the SE period (TH Tse), then few atoms undergo a SE collision during
the time when such a collision could lead to relaxation. For T = 145 C, TSE ≈ 20 µs, requiring a
minimum pulse amplitude B1 d × 1 µT. This is not a particularly stringent requirement on the
pulse height, though the optimum performance is achieved for pulses approaching a delta-function.
135
Figure 6.13: Simulation of the polarization components for Ω1T = 2πq × 1.25. The primary feature isΩz 6= 0 during the long part of the pulse cycle, so the polarization continues to precess. The resonances arestill prominent.
Hence SE relaxation is suppressed during the entire pulse cycle, despite the average magnetic
field Ωz over one cycle being arbitrarily large. While the transverse polarization is very sensitive
to the resonance condition, the pulse area is not as critical. As long as the pulse height satisfies
the description of the previous paragraph, SE relaxation is suppressed during the high part of the
pulse. For the low part, the choosing a ∼ 2π pulse ensures the cancellation of the the ambient
Ωz by the AC coupled field for resonant pulse frequency. SERF is achieved in low ambient fields
under the same conditions described in section 2.1.2, so the prescription to maintain SERF sen-
sitivity is TSEΩz(1 − x) 2π. Figure 6.13 shows a plot of the motion of the two components
of the polarization, when the pulse-width is not optimal. The most noticeable effect is that during
the time between pulses, the atoms continue to slowly precess, due to the imperfectly cancelled
ambient field. The resonances are still clearly visible, implying that optimization can be done
experimentally on the pulse height.
We have obtained some initial results for the transverse pumping, shown in figure 6.14. The
measured signal is the low-passed Faraday rotation measurement, which is proportional to Py. For
technical reasons, the plots were taken by setting ωz and Ω1 ≈ 2π/TH, and then scanning Ωz to find
136
the resonances. The bottom (blue) curve was taken by optimizing the pulse amplitude to minimize
all but the first resonance in the signal. The bottom and top units are related through Ω = γB/q.
For this measurement ωz = 2π × 35 kHz.
One indication that no SE relaxation occurs during the peak of the pulse is whether the pre-
cession occurs at a rate governed by q = 2I + 1. During the large part of the pulse, atoms in
different hyperfine manifolds precess in opposite directions at nearly the same frequency. If no
spin exchange collisions occur, precession in each hyperfine level occurs independently at the rate
given for q = 4, the free atom precession frequency. If SE collisions occur during precession, the
precession rate is slowed, as atoms spend some in each hyperfine level during precession. This
initial measurement suggests q = 3.5, which is probably off due to imperfect magnetic field coil
calibration.
To use the system as a magnetometer, one may apply a large field Ωz and feedback ω1 to
maintain resonance in the presence of an ambient field adding to Ωz. An experimental challenge in
this initial measurement was ensuring the pulse field and z-fields were uniform across all the atoms
in the cell. If they are not, then the resonance frequency varies across the cell, and broadens the
linewidth. A special set of coils was used for the pulse coils to help with this condition, described
in chapter G.
In addition to magnetometry, transverse pumping may be used in a novel form of spin-exchange
optical pumping (SEOP). Noble gas atoms may be polarized perpendicular to a magnetic field,
with similar physics to the description above, only with the alkali spin as the pumping term. The
noble gas precession frequency is set by the nuclear gyromagnetic ratio, and is around 1000 times
less than the rubidium gyromagnetic ratio. For successful SEOP, we require the alkali spins to be
aligned with the helium spins, and so we must be able to control the alkali precession.
The transverse pumping method descried above allows alignment along the pump direction that
is static compared to the timescale of noble gas precession. If the alkali pumping rate is modulate
R → Reiθ, where θ is the slow noble gas precession phase, then we have satisfied the necessary
requirements. Such a configuration could be useful in the construction of noble gas gyros.
137
Figure 6.14: Low-passed response of the transverse pumped SERF magnetometer, as the large DC-fieldΩ0 is swept. The data shows the relative insensitivity to the 2π-pulsewidth compared to the resonancelinewidth. The main effect from incorrect pulse area is the appearance of extra resonance features (redcurves). The blue curve shows the result for optimizing the pulsewidth by minimizing the DC resonancepeak. The extra resonances largely disappear.
138
Chapter 7
Biomagnetism
In this chapter, we focus on the particulars of the signals for which we are developing the
SERF magnetometer array: biomagnetic fetal MCG. We will talk about the general features of the
biomagnetic signals for which we intend our array to be suitable. We will demonstrate the signal
quality achievable by our array in real MCG applications, such as MCG and fMCG. Finally, we
will broach the general issue of biomagnetic signal processing. We do not pretend to provide an
exhaustive overview of this vast topic, aspiring merely to paint a picture of the applicable tech-
niques and limitations, and especially to discuss how they inform design and operation decisions
for our device.
7.1 MCG characteristics
We begin by examining the features of typical MCG signals. Databases of electrocardiogram
(ECG) signals exist in downloadable form for testing signal processing algorithms, such as the
PhysioBank ATM [Goldberger et al., 2000]. We use the ECGSYN, a resource from the PhysioNet
Toolkit, which uses a mathematical algorithm to generate realistic ECG waveforms [McSharry
et al., 2003]. The generator produces a digital waveform, which can be applied through the FPGA
139
output to a small coil or phantom head to test the magnetometer. The generator allows user selec-
tion of parameters such as heart rate and variation, QRS morphology and various other aspects of
the MCG signal, as well as additive noise.
The standard features of an MCG signal are in the lower-left inset of figure 7.1. During fMCG
arrhythmia detection, the desired information is related to the beat-to-beat timing interval of MCG
features, such as the QT or PR intervals. Tables for healthy fetal populations of different gestational
ages have been compiled, e.g. [Golbach et al., 2000], and deviations from these values can indicate
specific abnormalities.
For an adult MCG, we use a model of default ECGSYN parameters but minimized variation
and noise, at 60 bpm. We model a fetus with the same parameters, but at 130 bpm [Leeuwen et al.,
2004]. Figure 7.1 shows an example of the waveforms (inset), as well as the amplitude spectral
density for those waveforms. It is apparent that the region below 70 Hz is most important for
fMCG.
Another useful way of characterizing a typical MCG spectrum is with a joint time-frequency
transform. The Fourier spectrum of the MCG is itself periodic in time. Take a finite segment of
MCG data, such as the sample data in figure 7.1. We break this into i segments of length T ≤ Tbeat,
where Tbeat is the beat period (RR interval). The Fourier transform of each segment is the same
only when T = Tbeat.
There are many joint time-frequency techniques to examine this behavior. The short term
Fourier transform (STFT) is one such technique, which uses a windowed FFT, where the window
is time shifted to obtain the FFT as a function of time. Figure 7.2 shows an example of the STFT
applied to the sample MCG in figure 7.1, where the color scale is a log scale. The important
features of this figure are the widely varying FFT with time. Additionally, the P and T features
of the signal have low bandwidth, and to accurately identify where they begin for timing purposes
will require high sensitivity form 1-10 Hz for MCG, and at slightly higher frequencies for fMCG.
140
Figure 7.1: Simulated example MCG (black) and fMCG (red), from a mathematical model [McSharry et al.,2003] and generated by the online PhysioNet Toolkit ECGSYN waveform generator. The only differencein the simulated parameters are the average heart rate, 60 bpm for the MCG and 130 bpm for fMCG. Theamplitude for both signals is arbitrary. The upper right inset is a portion of the time series generated by theECGSYN generator. The lower left inset has the standard labelling for the features of an MCG
141
Figure 7.2: STFT of the sample MCG signal presented in figure 7.1. The STFT essentially uses a slidingwindow across the time domain to compute an FFT as a function of time. The color scale is logarithmic insignal size. The P, QRS-complex, and T waves are all visible. One important point of this figure is that toresolve the P and T waves, the frequency performance of the magnetometer below 10 Hz is important.
142
7.2 Signal processing
After the calibration and acquisition procedure (section 4.3), magnetometer recordings are cor-
rected by deconvolution with the magnetometer response. The DAQ program currently does this
by transforming to the frequency domain and then algebraically performing the deconvolution.
Specifically, if ξ(ω) = χ(ω)eiφ(ω), then B(t) is recovered by
B(ω) =Vm(ω)
ξ(ω)(7.1)
B(t) = IFFT(B(ω)
), (7.2)
where x(ω) = FFT (x(t)) and the m-subscript indicates the quantity from the originally measured
signal.
This initial step is completed on all signals in order to get a sensible measure of the magnetic
field during the recording time window. If the goal is to measure a signal of interest, in our case
a biomagnetic signal, further signal processing techniques are often available. Any information
that allows separation of intended signal and interfering noise is an avenue for signal processing.
The techniques we use fall into two categories: those which can be performed independently on a
single magnetometer measurement, and those requiring multiple magnetometers.
When we refer to noise, we are dealing with several different sources. Actual magnetic noise
adds to the signal of interest and would be detected by any magnetometer. SERF detector noise,
either fundamental or technical, also adds to each channel, but with different multichannel results.
Table 7.1 gives a sample of the characteristics of the various sources of signal that eventually makes
it to each magnetometer channel. Table 7.1 provides insight into how each noise source might be
distinguished from the sought signal, because different signal processing techniques are suitable
for SNR enhancement in the presence of different types of noise. In what follows, we will assume
we are attempting to detect an fMCG signal in the presence of all the noise sources listed in table
7.1.
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Table 7.1: Properties of noise sources for our magnetometers. For the spatial characteristics, “near” or“far” indicates effective magnetic source distance compared to the channel spacing. White noise impliesthe power spectral density is uniform and stationary. A signal localized in both the frequency and timedomain has regularly changing frequency content with time. Correlated indicates the technical meaningand does imply equality of response from all channels to the noise source. We use the nonuniform spatialcharacteristic label to indicate the noise source emanates from the detectors, and is generally differentfrom one detector to another. The detection noise includes electronic noise and noise relating to the probedetection scheme but unrelated to the intensity or frequency of the probe laser itself. An example would bebeam deflection on photodiodes due to vibrations.
Source Type Frequency/time rep. Spatial characteristics Correlated?
MCG/fMCG Magnetic Localized (both) Nearby source Yes
Coil circuit noise Magnetic white nonuniform No
HVAC fan Magnetic Single frequency Distant source Yes
Johnson noise (from MSR) Magnetic White Mostly far, uniform field Yes
Mother’s MCG Magnetic Localized (both) Further, not far Yes
Laser fluctuation Technical 1/f nonuniform Yes
Detection noise Technical 1/f nonuniform No
Quantum Fundamental white nonuniform No
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7.2.1 Single-channel processing techniques
Single channel signal processing relies on finding ways to eliminate parts of the signal that can-
not originate from fMCG, where signal processing can be applied to each magnetometer channel
separately. The lowest hanging fruit, here and in almost any application, is bandwidth limiting.
An accurate representation of an fMCG requires a bandwidth of about DC–80 Hz, as can be seen
from the sample fMCG in figure 7.1 (real measurements are abundant in the literature, e.g. Rassi
and Lewis [1995]). Higher frequencies are only noise and should be eliminated. This is done both
by hardware antialiasing filters (2-pole, 300 Hz low-pass for each channel) and by digital filtering
after data acquisition.
The next step in sophistication utilizes the joint time-frequency representation of fMCG. In this
representation, the frequencies contained in the signal are themselves periodic in time. Infinitely
periodic signals, such as the HVAC fan, are immediately identifiable as noise and can be separated
by their narrow frequency representation. These non-physiological signals could be removed by
notch filtering, however traditional FIR and IIR (types that obey causality) require compromises
in phase response leading to ringing and signal distortion as a price for high rejection of narrow
frequency. This can be a large problem for biomagnetic applications, where ringing distorts the
signal of interest [Leeuwen et al., 2004], and the ringing problem increases with the wavelength
selectivity of the filter.
A better approach to remove sinusoidal interference is direct subtraction. We use a LabView
VI (Extract single tone information.vi) to estimate the amplitude and phase of the sinusoidal signal
and then subtract it from the time domain signal. This technique works best for transform-limited
signals, i.e. when the signal bandwidth is limited by the FFT bin-width, rather than its own phase
noise. This method is easily implemented, does not introduce edge effects, and is effective at
eliminating sinusoidal noise sources without any ringing or overshoot.
The limited time-frequency content of fMCG can be used to further advantage. From figure
7.2, it is clear that higher frequency components of the MCG signal all occur within a narrow time-
window around the QRS peak. One way to use this information is through wavelet decomposition.
145
The basis of the technique can be described by an analog in data compression theory (indeed, noise
reduction and data compression are parallel fields).
The key idea is as follows: the Fourier transform is just one way of representing a function in
an infinite basis set, where the coefficients of that basis set are the amplitudes in the Fourier space
representation. Because the basis functions are sines and cosines, infinitely periodic functions have
a compact representation in this basis set, which is the same as saying a periodic function can be
well approximated by a few Fourier terms with amplitudes much larger than the others. If we had
reason to believe that our signal was composed of constant periodic functions, we could treat terms
with low FFT coefficients as noise without losing much information.
As figure 7.1 makes clear, MCG signals are only somewhat compactly represented in the fre-
quency domain. The amplitude of the Fourier terms drops off rather slowly from the fundamental
amplitude. This indicates it could be advantageous to choose a different set of basis functions.
Wavelet analysis uses limited time, limited frequency wavelets to express a function, and are a
common choice for the decomposition and filtering of biomagnetic signals [Khobragade, 1999,
Sternickel and Braginski, 2006].
7.2.2 Multiple-channel processing techniques
Methods in the previous section use function-space representations of the recording in which
the signal and/or interference have limited and separable representations. Frequency-space repre-
sentations can provide useful information on real-time or quasi-real-time frequency filters to use,
but more advanced techniques require time-intensive and manually tuned signal processing, and
currently must be completed offline. This represents a problem for real-time signal optimization,
when the signal of interested is comparable to background interference sources.
The spatial signature of a signal can be made to have different characteristics than the interfer-
ence, such as simply placing the detector as close as possible to the source and as far as possible
from interference. Using multiple, spatially separated magnetometers allows spatial filters to be
used, which can reject noise with different spatial frequencies than the signal. If the detectors are
146
close to the source, the spatial frequency will be relatively high, and therefore the noise spatial
frequency must be low in order for spatial filtering to increase SNR.
There are two situations to consider. The first is where the interference is primarily common-
mode to all magnetometers, as is the case for background magnetic field fluctuations from sources
far away from the magnetometers compared to the array spacing. Uniform fields correspond to
low spatial frequencies, and these are the signals best separated from fMCG.
Consider a noise source located far enough away that it can be represented as a dipole. Then
its field is
B(r) ∝ 3r (r ·m)−m
r3, (7.3)
where r is the vector from the center to the measurement point and m is the dipole generating the
field. Then we can write the fractional change in the field at two detectors separated by a distance
d as
∆B
B= 1− |B(r0 + d/2)|
|B(r0 − d/2)|
≈ (r0 − d/2)3
(r0 + d/2)3, (7.4)
where r0 is from the source to the center of the array and the approximation in the second line is
r0 d/2. For a noise source outside of the MSR case, r0 ≥ 2 m away from the array center, and
the array spacing 7 cm, the difference in signal between the two detectors is ∆B/B ≤ 10%. This
number is further reduced by the shielding factor of the MSR. Roughly the same analysis holds
true for Johnson noise from the MSR shielding, as long as the magnetometer array is centered
towards the middle of the room.
For the SNR of an MCG to be enhanced, it must have larger relative difference across the
magnetometer spacing than the background. An example of a magnetic field map from the heart
can be found in Steinhoff et al. [2004]. If the magnetometer array is centered well, two elements
should measure oppositely directed fields from the other pair, in which case ∆B/B > 1.
There is one other important source of magnetic interference in the fMCG application: the
mother’s signal. Figure 2 of [Steinhoff et al., 2004] displays the measured magnetic field map as a
function of position in the plane above the Thorax. At 12 cm above the chest and 30 cm away from
147
the center of the Thorax towards the naval, the magnetic field gradient of the QRS peak appears
to be approximately 0.4 pT/cm, so we would expect ∆B/B ≈ 2/3 from one magnetometer to
another, disregarding changes in the projection of the field vector on the magnetometer detection
direction over this region. Though this signal is obviously not strongly suppressed by detecting
gradients in the fields, and though the field cannot be described as common mode, it does have a
different spatial pattern than a fMCG if the magnetometers are placed in position above the fetal
heart. Further signal processing will be required to suppress the maternal interference.
7.2.3 Advanced nonlinear noise reduction
More complicated nonlinear state-space methods have been developed for MCG applications,
such as Sternickel et al. [2001], and used previously in atomic magnetometer biomagnetism exper-
iments [Weis and Wynands, 2005]. This algorithm works by using two channels, one as a signal
and one as a reference. We assume the signal channel contains the magnetic signal of interest,
deterministic magnetic background, and stochastic noise, while the reference channel contains the
deterministic magnetic background and uncorrelated stochastic noise. The nonlinear denoising al-
gorithm uses a state-space analysis to compare the signal and reference channels and discard the
correlated information to remove background magnetic noise. The filtered signal is then analyzed
again using a state-space method to suppress stochastic noise compared to the periodic fluctuations
of interest.
This technique still works best when narrow frequency components of the signal have been
removed first. The effect of this technique upon fetal signal extraction for clinical application is
still unclear, though it has been used for this application [Sternickel and Braginski, 2006]. An
example of this analysis on an adult MCG in a configuration where the magnetometers are spaced
far enough apart that only one is sensitive to MCG is shown in figure 7.3.
148
-8000
-6000
-4000
-2000
0
2000
4000
6000
Adu
lt M
CG
(fT
)
6543210Time (s)
(a) Raw MCG signal
-4000
-2000
0
2000
Adu
lt M
CG
Fie
ld (
fT)
543210Time (s)
(b) Nonlinear processed signal
Figure 7.3: Raw adult MCG signal (a) and nonlinear processed signal (b) using another reference channel.
149
7.2.4 Comparison of SERF and SQUIDs for noise processing
Signal processing biomagnetic signals is an active area of research. The relatively brief overview
here has only touched upon the most basic signal processing methods we have used. More ad-
vanced techniques can be found in the literature, and there is an open question how the signal
processing techniques traditionally developed for ECG or SQUID gradiometer signals should be
adapted to SERFs.
SERF magnetometers are capable of fantastic sensitivity. The fundamental noise limits of a
vapor cell with similar size to those used here is ∼50 aT/√
Hz [Dang et al., 2010], far below the
ambient magnetic noise level of a MSR. To take advantage of this extraordinary sensitivity in a
relatively noise MSR, multiple magnetometers must be used as gradiometers.
SQUIDs have been successful in developing hardware gradiometers for their detection cir-
cuits. In the typical configuration, SQUIDs are coupled to magnetic fields through superconduct-
ing pickup loops, which can be wrapped as two subtracting loops to make a hardware gradiometer
with high precision [Vrba and Robinson, 2001]. The equivalent task is more difficult for SERF
magnetometers. The easiest way to make gradiometers from a SERF magnetometer is to use a
single cell and detect the difference signal from spatially separate parts of the cell [Johnson et al.,
2010, Xia et al., 2006]. The advantage of this method is using one cell ensures the best uniformity
in the parameters of (2.24). For best SNR improvement, however, the cell would have to allow a
baseline separation of ∼ 5 cm or more if the signal from the source of interest is not to be sup-
pressed as well. This would require large vapor cells, which are problematic to heat, insulate, and
pump effectively.
In this work, we have pursued the goal of using multiple modular magnetometers to perform
the measurement. The obstacles to performing a gradiometric measurement are essentially non-
common mode noise in the magnetometer signal, either from real magnetic fields which are not
correlated from one magnetometer to another, or variations in the non-magnetic terms of (2.38) or
the spatial sensitivity profile of the atomic vapor, as in figure 5.5 and figure 6.2.
Uncorrelated magnetic noise can come from imperfectly correlated sources, mostly far away
(see previous section) or magnetic fields generated by our apparatus. One source of the later is
150
noise in the circuitry of the nulling coils. This applies a (mostly) white noise field to each magne-
tometer, independent of the others, which looks like a high spatial frequency signal to multichannel
processing methods. Another source is from the electrically operated heaters or any residual mag-
netization in the fiber components. Chapter 5 has more details about these sources.
Uncorrelated non-magnetic noise is especially problematic in the low frequency range from
0–10 Hz, still of interest in the MCG signal in figure 7.1. This noise is mainly the result of probe
beam fluctuations, and can be mitigated by modulating the signal and using lock-in detection tech-
niques [Li et al., 2006, Shah and Romalis, 2009a], as well as decreasing the overall probe intensity
(see figure 4.4). Another difficulty is that each magnetometer is sensitive to slightly different vec-
tor combinations of the ambient magnetic fields (see section 2.2.6), preventing straight-forward
subtraction of signals from one another, even after calibration.
Finally, our ability to accurately characterize the magnetometer limits our ability to make a
broadband gradiometer. The main problem is uncertainty in the frequency response of the mag-
netometer, seen in figure 4.1. The suppression of gradients is limited by the accuracy of this
characterization. Further, displaying a calibrated magnetic field time series is difficult to do in real
time. Feedback helps these issues.
7.3 MCG
To test the performance of our magnetometer and signal processing techniques, we used the
four channel array in the MSR to measure adult MCG. The procedure for obtaining these mea-
surements is described in section 4.5. We performed these measurements on 17 healthy adult with
informed consent. Adult MCG was easy to see even on top of the omnipresent fan noise.
Data representative of the obtained signal quality is shown in figure 7.4, where simultaneous
measurements from all four channels operating in DC mode are shown. The data has been filtered
to 128 Hz by a high tap-number FIR filter and detrended by a wavelet detrending technique. Also,
any sinusoidal interference identifiable in the noise spectrum was removed through the subtraction
method described in this chapter. The right-hand side of figure 7.4 shows average of about 30
beats. The data in figure 7.4(a),(b) are from different subjects. In (a), the array was positioned
151
over the subject’s chest where we estimated the signal should be maximized. In (b), the real time
MCG signal was used to guide the positioning of the array, which was adjusted until the signal
from all four channels was roughly equal. The P, QRS, and T components are well resolved.
We emphasize these are magnetometer signals (rather than gradiometer signals), and contain any
unfiltered background magnetic noise in the MSR as well as the MCG signal.
The data taken in figure 7.4(b) has an instructive noise spectrum from a signal processing
standpoint. Figure 7.5(a) shows the noise spectrum of a reference measurement taken immediately
after the measurement in figure 7.4(b), but without a subject in the MSR. The large, equally spaced,
and narrow set of frequency components was later identified as arising from the electronics for a
permanently installed SQUID system (Tristan Technologies Inc) in the MSR. Since it is composed
of narrowly peaked frequency components, we deal with it as described earlier in this chapter.
Sinusoidal functions were matched to each frequency component and subtracted from the time
series. The results after this process and the usual detrending are shown for both the MCG signal
and reference, in figure 7.5(b).
152
(a) MCG, uncentered array
(b) MCG, centered array
Figure 7.4: MCG acquired from different subjects, representative of the signal quality obtained in all runs.The lefthand side of each figure shows the signal obtained after subtraction of sinusoidal interference andthe application of a detrending algorithm. The righthand side shows about 30 averaged beats. In (a), themagnetometers were positioned above the chest by sight. In (b), the real-time signal was used to guide thepositioning of the magnetometers to maximize the signal in all channels.
153
Figure 7.5: (a) Reference noise spectrum taken immediately after the data presented in figure 7.4(b). Theequally spaced peaks are from electronics for one of the SQUID systems permanently installed in the MSR.This noise was present during the MCG measurement. (b) power spectral density of the filtered MCGmeasurement (black) and the filtered reference measurement (red dots). Each frequency component wasremoved by finding and removing a sinusoidal signal from the time series. The noise was effectivelysuppressed.
154
7.4 Phantom measurements
To test the magnetometer measurement capabilities for signals of similar size, bandwidth, and
spatial structure to fMCG, we also simulated fMCG using a head phantom (Biomagnetic Tech-
nologies), which is a spherical shell filled with saline solution and five different current dipoles
that can be driven individually. The head was placed as close as possible to the array center and
aligned so that the detected signal from driving a dipole was maximized. We drove the phantom
with a test waveform [Goldberger et al., 2000] shown in figure 7.6(b). The amplitude was varied
by changing the voltage scaling of the signal applied to the phantom head driver, and the calibrated
magnetometers were used to adjust the magnetic field at the detectors to the desired level.
Figure 7.6(a) shows 28-beat averages for all four magnetometer channels, after processing the
signals in the same way as the MCGs of the previous section. The detected QRS amplitude is about
2 pT, and the averages for each channel have the correct general shape and representation.
155
Figure 7.6: (a) 28-beat average response from the phantom driven with the waveform shown in (b), withmagnetic field offsets applied to aid visualization. Before averaging, the signals are processed in the sameway as the MCG above, by removing only narrow frequency components.
156
7.5 Fetal MCG
The goal of this work was to make a device capable of fMCG detection and to investigate
the suitability of such a device to replace SQUIDs for some biomagnetism applications. We have
now tested one healthy fMCG subject, at 31 weeks gestation. For most of the measurements here,
the ambient noise sources are minimal. The HVAC fan was shut down and unnecessary SQUID
electronics were turned off during our portion of the measurement.
Figure 7.7 shows the raw data as it looks before calibration and more sophisticated signal
processing. Figure 7.7(a) shows the large common mode baseline drift that often occurs. The
drift amplitude in this figure is actually relatively small. It is mostly unimportant until it saturates
the I-V converter for a channel, and then it is catastrophic. For comparison, the I-V maximum
output is ±6 V. The large peaks in every channel are the mother’s MCG signal. The right hand
side of Figure 7.7(a) displays a blowup portion of channel 2, which seemed to have the largest
amplitude fMCG signal. The fMCG QRS-complex is easily seen (highlighted in red) without any
signal processing (except for the 300 Hz anti-aliasing filter applied by the I-V converters). The
first fMCG QRS-complex in this selection is visible, but is mostly hidden by the mother’s MCG, a
common challenge for fMCG and illustrative of the need for multichannel techniques, where these
features can be isolated by spatial filtering.
Figure 7.7(b) shows the response from the raw signal in (a) bandpass filtered from 3-30 Hz
using two single pole filters. These filters are available on demand from the I-V converters. The
3 Hz high-pass is sufficient to remove the large baseline drift, and the left-hand side of (b) empha-
sizes the largely uniform MCG signal from the mother. A blowup of these signals (right-hand side
of (b)), with an offset applied to separate them, shows the portion of the signal that contains the
fMCG QRS complex, picked out most easily from channel-2 but present in all channels. This is an
important feature if the array position is to be optimized to get the largest signal.
Figure 7.8 shows the same set of signals after calibration, detrending, and removal of of sharply
peaked frequency components. Additionally, the signal was down sampled to 200 Hz. A time-
segment with different offsets applied for channels 1-2 and channels 3-4 highlights the close QRS
157
Figure 7.7: (a) Raw, uncalibrated fMCG signals. The inset on the right hand side shows the fMCG isclearly identifiable in the raw signal without any signal processing. Fetal MCG beats are highlighted withred ellipses. (b) Bandpass filtered signals from (a), at 3-30 Hz, chosen to enhance the QRS SNR for realtime visualization and optimization. The right hand side shows the signals on a smaller scale with offsetsapplied. The signal is visible in each channel.
158
Figure 7.8: The calibrated signal filtered to remove 40 Hz and 60 Hz harmonics, low-passed 200 Hz usinga FIR filter, and detrended to remove low-frequency noise. The mother’s QRS in Channels 1-2 are wellmatched, and 3-4 are well matched, the amplitude is not uniform between the two sets. A short a short timesegment of this data is shown, where differences in the fMCG QRS complex phase are visible.
159
peak agreement among those subsets. On the right of figure 7.8 is an even shorter segment from
the same dataset, where the phase difference between the fCMG QRS components is apparent. To
reiterate, these images show the mother’s QRS is the same, only amplitude scaled, while the fetal
QRS has an entirely different profile for the different channels.
To remove the mother’s signal, we use principal component analysis (PCA), in which the cross-
correlation matrix of the signals is calculated and diagonalized to produce the principal compo-
nents. The first principal component corresponds to the part of all four signals that is most strongly
correlated, and simple inspection of figure 7.8 indicates that should be the mother’s signal and any
remaining common mode background, as well as a small amount of fetal signal. Deleting this
component and then reconstructing the signal without it offers a crude way of eliminating com-
mon mode signal through correlation, rather than direct subtraction. The results of this procedure
performed on the data in figure 7.8 are shown in figure 7.9. The mother’s signal has been almost
entirely suppressed, and the fetal signal is easily visible.
Appendix F contains a report generated by Ron Wakai’s group from the set of measurements
obtained with this first subject. The report computes clinically relevant fMCG parameters, such as
the QT, PR, and RR intervals, as well as the P and QRS duration. Two different segments of a single
data set were analyzed, and the final page contains the same measurements made by a commercial
SQUID system that is normally used for such measurements. The SERF array measurements agree
reasonably well with the SQUID data, and would obviously benefit from the introduction of 4-more
channels by using z-mode. The next step for the biomagnetism portion of this experiment are to
use z-mode to measure 8-channel data, and to replicate these exciting results with more subjects.
160
Figure 7.9: Data with the first principal component removed, as discussed in text. The mother’s signal hasbeen suppressed, and the fMCG QRS complex is the largest remaining feature.
161
Chapter 8
Conclusions and future work
In this thesis, we have presented the design and implementation of a four channel magnetome-
ter array. We offered a complete description of the array, its noise sources, and its theory and
method of operation. Additionally, we discussed different operating modes designed to overcome
shortcomings for SERF, including using feedback to make a gradiometer, utilizing parametric mod-
ulation modes to increase the channel count or operate in high DC bias fields, or take advantage of
diffusion to make magnetometer measurements in a part of the cell without pump light. Finally,
we proved for the first time that an atomic magnetometer array can be used to detect fMCG, a
demanding biomagnetic application.
There are two general directions for future work. First, fMCG measurements must be replicated
and improved. The gestation age of the fMCG in this work was relatively late, 31-weeks. Judging
by the signal quality in this work and the way fMCG signals develop [Golbach et al., 2000], earlier
detection should be possible. Additionally, an effort should be made to better match the feedback
field to all magnetometers. Finally, z-mode operation needs to be attempted to increase the channel
count and see if signal processing algorithms correspondingly benefit.
162
Longer term, there are several improvements that can be made. The current magnetometer
design philosophy pays a space premium in exchange for optical flexibility. In the future, mag-
netometer size can be reduced further by sacrificing this flexibility in place of permanent optical
setups, which would allow increasing the channel count.
A new setup might use diffusion mode pumping and small grin-lens terminated fibers. The
optical components could be glued directly to the lenses and attached to the cell or to a thermally
insulating spacer. To significantly shrink the magnetometer, a beam cube (rather than a Wollaston)
should be investigated, with the detection photodiodes glued directly to the cube faces.
Another improvement is background suppression of the ambient magnetic fields in the MSR.
The feedback technique discussed here works well as long as the magnetic fields do not change
significantly in the directions orthogonal to the feedback direction. When they do, the DC fields
can acquire signficant offsets that require re-nulling. During especially noise periods, this can
happen often enough that measurement becomes difficult, as the signals become saturated during
recording. The only solution to this problem is feedback stabilization of the background fields in
all directions. This could be done with a set of semi-permanent coils wrapped around the walls
of the MSR and a single reference magnetometer in the middle, operating in the feedback method
described in Seltzer and Romalis [2004], whose stabilization levels were more than sufficient for
work in our MSR.
Finally, there might be significant benefit using hybrid magnetometer cells, with a dual alkali
optical pumping technique. For example, Rubidium could be optically pumped and used to pump
Potassium intermixed in the cell via spin-exchange. Several benefits come from the Potassium not
interacting with the pump laser, such as reduced light shift and laser absorption at the cell walls.
The comparison fMCG measurement of our SERF magnetometer to the commercial SQUID
measurement of an fMCG signal shows both that SERF magnetometers have a long way to go, but
also that they have come a long way. We must remember the technology will only be ten years old
this year. Considering the results presented in this work, the SERF magnetometer promises contin-
ued improvement in biomagnetic applications as more technical advances are made. If successful,
163
they will provide a cheaper and less maintenance intensive alternative to SQUIDs, and may enable
much smaller cylindrical shields rather than expensive magnetically shielded rooms.
164
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172
Appendix A: Dipole electric polarizability
The dipole polarizability tensor is a quantity that fundamentally describes the interaction be-
tween atoms and electric fields, including those created by light. Its usefulness derives from the
ability to separate the angular momentum properties of transitions from a characteristic coupling
strength that depends upon atomic details, and to compress the atomic details into the oscillator
strength. We will follow the framework of [Bonin and Kresin, 1997] to express the atomic polar-
izability for 87Rb in the fine structure basis assuming unresolved hyperfine.
A.1 Fine structure polarizability
As a polarizable medium, when a laser field propagates through an atomic medium, the first
order response is an oscillating dipole, with which the propagating field also interacts. This behav-
ior is characterized by the polarizability tensor operator α, and in the presence of an oscillating
electric field, leads to a density matrix evolution characterized by the Liouville equation
i~dρ
dt= [H0, ρ] + [δH, ρ]. (A.1)
The polarizability can be expressed conveniently in a spherical-tensor basis set [Bonin and Kresin,
1997],
αij = α0δij − α1iεijkJk + α2
32(JiJj + JjJi)− J(J + 1)δij
J(2J − 1), (A.2)
where δij = δij I , where the right hand side is the usual Kronecker-delta function and the identity
operator, respectively. The angular momentum dependence of the polarizability is written explic-
itly in (A.2). The coefficients are written
α0 =∑l 6=0
Al0G+(l, 0, ω)φ0(J, Jl) (A.3)
α1 = 2∑l 6=0
Al0G−(l, 0, ω)φ1(J, Jl) (A.4)
α2 =∑l 6=0
Al0G+(l, 0, ω)φ2(J, Jl). (A.5)
173
The sums are over all atomic levels l not equal to the shifted level k. The various quantities used
to calculate these coefficients are
G±(ω) =1
ωl0 − ω − i∆Γl/2± 1
ωl0 + ω + i∆Γl/2(A.6)
φ0(J, Jl) = δJl,J−1 + δJl,J + δJl,J+1 (A.7)
φ0(J, Jl) = − 1
JδJl,J−1 −
1
J(J + 1)δJl,J +
1
J + 1δJl,J+1 (A.8)
φ0(J, Jl) = −δJl,J−1 +2J − 1
J + 1δJl,J −
J(2J − 1)
(J + 1)(2J + 3)δJl,J+1 (A.9)
Al0 =rec
2
2ωl0fl0, (A.10)
where ωl0 is the transition frequency between the shifted level and the level denoted by quantum
numbers l, and the oscillator strength between the two levels is fl0.
A.2 Light Shift
The atomic polarizability has a vector component that gives rise to an effective magnetic field
[Bonin and Kresin, 1997, Happer and Mathur, 1967, Kornack, 2005]. Continuing to use the for-
malism of Bonin and Kresin [1997], for a pump laser with narrow linewidth and near resonant
with the D1 line, only one excited state in the sum of (A.3)–(A.5) participates, J = Jl = 1/2, and
we make the rotating wave approximation to simplify G±. Under these assumptions, the tensor
polarizability for the fine-structure ground state 52S1/2 coupled only to the 52P1/2 state is
αij = AD1G(ω)(δij + 2iεijkJk) (A.11)
G(ω) =∆ω + i∆Γ/2
∆ω2 + (∆Γ/2)(A.12)
The perturbing term for an oscillating field is
δH = −|E|2
4ε∗ · α · ε (A.13)
where ε is the polarization vector for the electric field. The real part of δH corresponds to disper-
sion, while the imaginary part corresponds to absorption, and the dispersive part will be responsible
174
for the effective field from the light shift. For our magnetometer, we are after the energy shift
〈δE〉 = Tr(ρ<(δH)), (A.14)
which will be an effective magnetic field because we will write it in the form
〈δE〉 = gsµBBLS ·P, (A.15)
where BLS will be the effective magnetic field seen because of the light shift.
To evaluate (A.14), we need to choose a laser polarization. For the Cartesian basis, we express
the polarization in terms of the circular phase θ,
ε =
cos(θ)
i sin(θ)
0
eiαx . (A.16)
Following the convention of [Shah and Romalis, 2009b], the polarization vector s = ε∗×εi
=
sin(2θ)z. Then
<(δH) = −|E|2
4Al0(I − 2 sin(2θ)Jz)<(G(ω)) (A.17)
<(δH) = −|E|2
4Al0(I − s · 2S)<(G(ω)). (A.18)
Finally, we can use an explicit matrix form to evaluate the trace in (A.14), with
I =
1 0
0 1
, 2Jz =
1 0
0 −1
, ρ =
1 + P 0
0 1− P
. (A.19)
The result for the energy shift is
〈δE〉 = −~refcΦ0D(ν)(1− s ·P), (A.20)
where we have substituted
|E|2 =I
nε0c=
8πI
c, (A.21)
and
D(ν) =1
2π<(G(ω)) =
1
2π
∆ν
∆ν2 + ∆Γ/22 (A.22)
175
and Φ0 = I/hν0 is the photon flux. We can rewrite (A.20) by inspection in the form of (A.15) to
get the effective field caused by the light shift (ignoring the constant shift to all levels),
ΩLS = refcΦ0D(ν)z, (A.23)
which agrees with equation (9) of [Shah and Romalis, 2009b]. In the fine-structure basis, we can
treat this field precisely as any other Ωz, and the plots in chapter 2 remain the same. In reality, the
Gaussian spatial dependence of the pump laser causing the light shift, as well as the absorption of
the pump laser as it passes through the cell, make the light shift a complicated function of position.
In our experiments, the measured Ωz is a combination of real fields and the light shift field, and
its value is minimized but not zeroed during the nulling procedure.
A.3 Faraday rotation
The measurement of atomic spins in our system utilizes the polarization rotation of a far off
resonance probe beam. For the D2 transition with unresolved hyperfine structure and in the rotating
wave approximation, the polarizability is
αij|D2 =AD2
2π(L(∆ν) + iD(∆ν)) (δij − iεijkJk). (A.24)
where
A =refc
2
2ω0
(A.25)
δij = δij I (A.26)
D(∆ν) =∆ν
∆ν2 + (∆Γ/2)2. (A.27)
Equation (A.24) is written in an irreducible spherical basis set, with scalar and vector components
(the tensor component is zero)
αij = αscδij + αvij
αsc =1
3αii
αvij =1
2(αij − αji) .
176
If we use the geometry in figure 2.6, the polarization of the probe beam is in the y − z plane. The
vector component of (A.24) that will rotate the probe polarization in this plane is given by
αvyz = − i2
A
2π(L(∆ν) + iD(∆ν)) Jx (A.28)
Rotation can be described in terms of the index of refraction nr, which is related to the polarizabil-
ity through Kramers-Kronig relations [Bonin and Kresin, 1997],
nr = 1− 2π[Rb]R(α). (A.29)
We are interested in the phase rotation of the probe beam versus the Px = 0 case. This angle is
φ = ∆nrωt (A.30)
∆nr = 2π[Rb]R(Tr(ραvyz)), (A.31)
with the propagation time t ≈ l/c for small changes in the index of refraction. The resulting
expression is
φ = −1
4[Rb]lrefD2cD(∆ν)Px, (A.32)
which agrees with [Ledbetter et al., 2008] eq. 13.
If rotation from the D1 line is important (for example, in high pressure buffer gas cells), it can
easily be incorporated. The D1 polarizability has the same form as (A.24), with some different
numerical factors,
αij|D1 =AD1
2π(L(∆ν) + iD(∆ν)) (δij + 2iεijkJk). (A.33)
Comparison of (A.33) to (A.24) allows us to write down the rotation from the D1 line as opposite
in direction and the same numerical factor aside from the frequency dependence (fD2 = 2fD1
cancels the extra factor of 2 in rotation), and the total rotation for a given detuning is
φ =1
4[Rb]lrefD2c (−D(∆νD1) +D(∆νD2)) . (A.34)
177
Appendix B: Operating mode parameters
This appendix is designed to provide a reference for the parameters used for particular operat-
ing modes and in our calculations of derived quantities. Daily readjustment of the magnetometers
by response optimization makes these parameters only estimates of common operating conditions
for each magnetometer. Further, cell-to-cell variation (appendix C) makes discussions of calcula-
tions of derived quantities more qualitative than quantitative.
B.1 Standard parameters for all calculations
Symbol Value Units Description
Tc 145 C Cell temperaturePN2 50 (0.066) Torr (Amg) N2 pressure at 300 CPHe 22 (0.029) Torr (Amg) He pressurel 1 cm Cell side length (modelled as a cube)
178
B.2 DC-Mode
(a) Pump optics for DC mode (b) Probe optics for DC mode
Figure B.1: DC-mode optics
Symbol Value Units Description
Ppu 4 mW Pump powerwpu 0.256 cm Pump Waist
∆ν1, (x1) 20(5.9) GHz (linewidths) Pump D1 detuningPpr 1. mW Probe powerwpr 0.11 cm Probe Waist
∆ν2, (x2) −98.5(−28.9) GHz (Linewidths) Probe D2 detuning
179
B.3 Diffusion pumped mode
(a) Pump optics for diffusion pumped mode (b) Probe optics for diffusion pumped mode
Figure B.2: Diffusion pumped optics
Symbol Value Units Description
Ppu 4 mW Pump powerwpu 0.051 cm Pump Waist
∆ν1, (x1) 20(5.9) GHz (linewidths) Pump D1 detuningPpr 1. mW Probe powerwpr 0.27 cm Probe Waist
∆ν2, (x2) −98.5(−28.9) GHz (Linewidths) Probe D2 detuning
180
Appendix C: Cell characterizations
This appendix has some standard measurements for each of the cells. Though three of the four
have the same specifications, they are all different. In this appendix, we try to give a feeling for the
individual performance of the four cells used throughout this experiment.
Table C.1 describes the initial nominal properties of the cells. We have some reason for dis-
trusting the cells from Triad Technologies, as cells were initially sent with no N2 buffer gas, or
with incorrect sizing, or mysteriously, seemingly without Rb. Initial measurements confirmed the
total buffergas pressure, at any rate, for the cells used.
As mentioned in the text, He diffuses through Pyrex on timescales not insignificant over the
course of the measurements presented in this work. We obtained the cells from Triad Technologies
in mid-march, 2010. Figure C.1 shows the expected time dependence for diffusion loss of the He
through the Pyrex.
Table C.1: The initial nominal properties of each cell. Cells 1–4 are used in all magnetometry experimentsdescribed here, while the oscillator cell was used in the transverse pumping experiment. All cells areconstructed of Pyrex, and have dimensions 1 x 1 x∼5 cm3.
Cell # Manufacturer Nominal N2 (Torr—amg) Nominal He (Torr—amg)
1 Triad Technologies 50 — 0.066 760 — 1
2 (like cell 1)
3 Opthos Instruments Inc. 100 — 0.132 –
4 (like cell 1)
Osc. Opthos Instruments Inc. 400 — 0.524 –
181
Figure C.1: Expected He diffusion from cells 1,2, and 4. The values roughly agree with measurementsextracted from pressure broadened linewidths measured over this time period.
Figure C.2: Uncalibrated sweep of By performed simultaneously for each magnetometer. The field isapplied with the large coil used for feedback gradiometry, so is nearly the same for each cell. While thedetails of the sweep shape depends on many individual parameters (section 2.2.5), general features arerepresented here. For example, cell 2 always has the smallest response, and cells 2-3 always seem to bebroader than the others. This may be due to cell birefringence causing probe ellipticity in these cells.
182
Appendix D: Materials
Many non-traditional materials were researched for this project. The primary requirement
for all materials near the magnetometer is that they be non-conducting to eliminate Johnson noise.
The thermal parameters are important for the cell, heatsink, and insulation; particularly the thermal
conductivity, coefficient of thermal expansion, and maximum use temperature. We have compiled
at list of useful materials and their properties in table D.1.
The optics tubes and mirror mounts are made from Delrin. The plastic mount for the mag-
netometer is made from ABS. The thermal insulation for the heatsink made from Aerogel. The
heat-sinks are made from Boron Nitride, though Shapal-M would have been a better choice. The
Boron Nitride heat-sink is clamped together with FEP tape, which is also used to press the Pyrex
cell against the heat-sink.
Several of the plastics used in this work are difficult to bond to each other or to other materials.
We have had success bonding Delrin to itself and to glass using Permabond Standard Super Glue
105 and 200. These glues are particularly useful for gluing plastic retaining rings to optics or
plastic optics adapters. The optic and ring are clamped together, then a small bead of glue applied
on the edge of the optic, against the retaining ring.
183
Table D.1: Thermal material properties for useful non-magnetic, non-conductive materials. Tmax is themaximum use temperature, k is the thermal conductivity and αL is the linear coefficient of thermal expan-sion.
Material Tmax(C) k (W/m K) αL (10−6 1/C) Notes
Macor 800 1.46 0.93Machinable, low ther-mal conductivity
Boron Nitride 550 34–66 0.7–1.3Machinable, higher ther-mal conductivity
Shapal-M 1000 90 4.4Machinable, highestthermal conductivity
ABS 88 0.17 50–85Good all around choice,can chemically weld
Delrin 85 – 123Good mechanical prop-erties, slippery and diffi-cult to bond
Peek 249 0.25 46.8Good at high temp., bet-ter mechanical proper-ties than PTFE
PTFE 250 0.23 100Good at high temp., onlysticks to itself
Pyrex 230 1 4Glass for vapor cells,will not hold He gas
Aerogel 200 0.022 –Good insulation, lowerthermal conductivitythan air
Polyamide 288 0.035 – Rigid foam insulation
FEP Adhesive tape 204 – –Plastic tape sticks toPTFE, high temp
184
Appendix E: Electronics
This appendix focuses on the electronics circuits designed and developed as part of the work
presented in this thesis.
E.1 Heater H-bridge circuit
The original prototyped magnetometer was driven with a set of heaters operated in the man-
ner of chapter 3, driven with a DC current. In this configuration, with dual heaters aligned to
cancel out single-heater magnetic fields, and heater pairs aligned to suppress remaining fields, the
0.5 A DC currents produce fields that are measurable, but smallest in the sensitive direction of the
magnetometer due to the heater orientation. Measurements indicate residual fields <1 nT in this
direction. No difference was noticed in the noise spectrum between when the heaters were on or
off.
Using the 4-channel magnetometer array with a single large set of tri-axial nulling coils, mag-
netic gradients from the heater fields became noticeable and reduced the magnetometer sensitivity.
The magnetic fields could still be nulled at each magnetometer individually, but it became impos-
sible to cancel the fields at every magnetometer simultaneously with four independent sources and
only a single control. To address this problem, we developed circuitry to drive the heaters at a high
enough frequency that fields caused by the heaters average out on typical biomagnetic timescales.
The circuit is an example of a commonly used technique in stepper-motor control and is
called an H-bridge. Functionally, it is simply a set of four digitally controlled MOSFET switches
which are controlled to periodically switch the current direction through the load by synchronously
switching the sides connected to the DC power supply and ground (see inset to figure E.1). The
full circuit schematic is shown in figure E.1. Many ICs exist to facilitate driving the MOSFETs
properly. We have chosen a single integrated IC, the ISL83204A, which has many useful features.
With inputs of +15 V, it can drive the high-side N-channel MOSFET gates to about V++10 volts.
It can also operate in self-oscillatory mode with an oscillation period defined by the value of a few
external components.
185
The circuit can be configured in one of two modes: operation between a positive rail and ground
(V+ mode) or operation between ground and a negative rail (V− mode), with only a small change
in the circuit hardware. Each magnetometer requires one PCB and a dedicated adjustable DC
power supply, along with +/- 15 V supplies for the ICs. A list of the circuit elements is provided in
table E.1.
The simplest setup is V+ mode, made for switching the heaters with a positive DC source. In
this case, R16 should be bypassed, R17, Q1, Q2, and U1 are unnecessary, and the indicated connec-
tion should be made across the pins for Q2. Connections for edge connector J1 are V+ = Vcontrol,
V- = GND.
In V+ mode, a variable DC voltage source provides V+. U2 provides a reference voltage
at 7.5 V, and the combination of C2, R3, R4, R5, and pots R10 and R11 combine to make a
relaxation oscillator with circuitry inside the ISL83204A to determine the gate control switching
frequency. The time constant is set by τ R5, R10series+R11series, and C2. Using C2=250 pF and
1 k≤R5+10+11 ≤2.11 M, the oscillation frequency can be adjusted from 1 kHz to 1 MHz.
J5 connects to a ON/OFF/ON DPDT front-panel switch which acts to turn the switching on,
high-only, or both sides of the load connected to ground. Resistors R8 and R9 provide dead-time
control of the switching circuit to eliminate both switches on a single side of the output load being
closed at once, a condition known as shoot-through, which becomes more problematic at higher
switching frequencies. D1,2 and C3,4 provide power for the bootstrapping circuit that ensures the
high-side gate control voltage is higher than V+. D3–6 and R12–15 are to lower pickup noise on
the gate control voltages, if necessary. We have not used them, but if noise in the circuit becomes
too large they may help. In this case, R12–15 should be 10 Ω, and the diodes should be fast
recovery types for the highest speed switching.
The actual switching is done by U5,6 which are integrated double MOSFETs arranged with
common source-drain connections to facilitate smaller H-bridge component count in situations
(such as this) where the switched power is not too high and the chips will not become too hot. J6
is a screw-terminal output connection that goes to the load, while J3 is a set of monitor pins that
can be connected to J4 (which leads to a differential amplifier) or can be monitored directly.
186
Care must be taken when monitoring the output. If coaxial cable is used to connect the circuit
output to the load, one must remember the power and grounds are quickly switching, so acciden-
tally grounding a shield for a coaxial cable (by connecting to a scope input, for example) will short
the circuit during 1/2 of its cycle.
V− mode works in essentially the same was as described above, except a virtual ground (V-)
is used, and the H-Bridge controller IC is operated between V− and 10 V+V− as power supplies.
All circuit elements are used (except the optional gate resistors and diodes). U1 is a low-drop 10 V
voltage regulator, chosen to ensure it works between the full range of 15 V-V−. The extra circuitry
is an adaptation of a circuit in [Horowitz and Hill, 1989] to increase the maximum operation current
at the regulated voltage. The input connections to J1 are V+=GND, V− is provided by a variable
voltage DC source.
In either operation mode, the output power to the heaters is controlled by adjusting the ampli-
tude of the DC power supply voltage. This circuit will work across the full range V+=|V−| ≤ 20 V
available from the programmable National Instruments PXI-4110 power supplies used in this work.
We have put four of these circuits in a electronics box. The front panel connections are the
IC power connections ±15 V and GND, as well as two positive and two negative DC voltages
which will control each of four magnetometers, and the ground for the DC power supplies. The
output is an 8P8C Jack, meant to connect to shielded-Cat6e cabling. Shielding the heater wires
is important, as the frequencies and currents are higher and pickup noise exacerbated. Pickup is
primarily a problem for the photodiode signals, where the large high frequency pickup is aliased
into the detection frequency band.
The higher frequency components of the square wave modulation seem to induce oscillations
on other parts of the magnetometer circuitry, most importantly the photodiode cables. There is
a ring-down with an oscillation frequency of around 12 MHz, which might correspond to the
transmission line behavior of the cables. While this frequency is much higher than those used
in magnetometry, they get aliased into the detection band, and for higher modulation frequencies
(∼ 500 kHz) the aliasing noise becomes noticeable. Lower modulation frequencies induce less
ring-down, but perturb the atoms more. While the current setup allows DC heating because there
187
is individual control over each magnetometer, sine-wave modulation should be considered in the
future, perhaps with integral PID control of heater currents.
188
Figure E.1: Heater H-bridge controller
189
Table E.1: Parts list for the heater controller. D3–D6 and R12–R15 are bypassed unless there’s excessivepickup noise in the switching circuit.
Comp. Value Notes Comp. Value
C1 22uf R1 10kC2 250p R2 10kC3 100n Good leakage/Rf properties R3 100kC4 100n Good leakage/Rf properties R4 100kC5 0.22u R5 2kC6 100u Only for V- circuits R6 10kC7 100n Only for V- circuits R7 10kC8 22u R8 100kC9 1u R9 100k
C10 1u R10 2M 1TC13 100n R11 100k 10TC16 10u R12C18 10u R13C19 10u R14D1 1N5817 R15D2 1N5817 V- only R16 3 1/2WD3 1N4448 V- only R17 1 2WD4 1N4448 R18 100kD5 1N4448 R19 100kD6 1N4448 R20 10kJ1 6-pin screw terminal Input power connections R21 10kJ2 2-pin crimp post Mon. Comparator (V- only) U1 TLE 4276J3 2-pin crimp post Mon. Bridge U2 LF411J4 2-pin crimp post Mon. Out U3 ISL83204AJ5 4-pin crimp post Dis/HEN/On U4 TL071J6 2-pin screw terminal H-Bridge Out U5 IRFI4019HG-117PJ7 2-pin crimp post Mon. Out U6 IRFI4019HG-117PQ1 2N5194 Only for V- circuitsQ2 2N5194 Only for V- circuits
190
E.2 Thermistors
Many thermistors contain magnetic materials, and many more cannot be used at the high tem-
peratures required for atomic magnetometry. Simply testing the thermistor (including the leads)
with a strong magnet is usually sufficient. We have used Honeywell NTC Thermistor 112-103FAJ-
B01 in the magnetometers presented in this work. Standard thermistor curves can be used to predict
the temperature from the resistance, given R0 = 10 kΩ and β = 3669. For reference, a plot of the
predicted temperature vs thermistor resistance for the thermistor in all channels is shown in figure
E.2.
Figure E.2: Nominal thermistor temperature vs. resistance for the thermistors used in all channels.
E.3 Coil driving circuit
An essential component to the magnetometer is the circuitry used to apply offset and calibration
fields. The coil control box built for this circuit allows a DC offset and input signal to be coupled
and are applied to the coils using a voltage to current converter. There are three characteristic
voltages in the box: the DC voltage selected with a front-panel potentiometer (Vdc), the input
voltage Vi, and the final output voltage Vo. The job of the coil control box is to make giVi+gdcVdc =
191
Table E.2: Coil control box design features
Front panel inputs
BNC for ViPotentiometer to adjust VdcSwitch to adjust gi =1,1/10,1/100AUX input
DIP Switches insideAdjust R =50,500,1k,5kAdjust gdc =1,1/10,1/100
OutputsBNC monitors VoRJ45 receptacle for an 8P8C Cat. 6 cable
FeaturesEach enclosure controlls 3 boardsUsing ±15 V supplies, Vo(max)=12 V
Vo, and to output Vo across a series resistor R as a current source. Additionally, it must perform
this function without adding too much current noise, which is converted to magnetic noise at the
coils. With the coil calibration ∼20 fT/nA, the maximum allowable current noise in the coils is
50 pA/√
Hz. This is on top of a driving current of up to 2 mA for DC cancellation.
The circuit presented here was designed as part of this work and built by Greg Smetana. A list
of features of the coil control box are described below:
The circuit diagram for the coil control box is shown in figure E.3. There are three primary
subcircuits. In the red-dashed outline, two LM399 Zener reference diodes are used to create a
stable reference on either side of ground. The wiper of a front-panel potentiometer selects the
DC voltage from ±Vzener and is buffered by an op-amp to produce Vdc. Resistor networks allow
division of this voltage in decade steps from 1–100. The blue-dotted line accepts a front panel
input voltage Vi using a low-noise differential amplifier, and a resistive voltage divider can step
down the voltage in three decade steps from 1–100. The final piece is a balanced line driver using
two input op-amps to combine the voltages from previous subcircuits. A third op-amp converts
the balanced line driver into a current source by monitoring the output voltage across one of the
resistors and feeding back to the input amp. The output resistors can be varied in values of 50, 500,
1k, and 5k. Resistive values much higher are not useful at our voltage ranges (i.e. typical Vo is a
few volts across 5k).
192
Figure E.3: Coil control box circuit diagram
193
The transverse oscillator has even more severe demands. The coil calibration is larger by a
factor of 100, while the necessary currents are larger by a factor of 100 also. Thus, the maximum
current for 1 fT/√
Hz noise level in the direction of the large field is 500 fA/√
Hz, on top of a
200 mA signal!
We have made measurements of the current noise for zero applied current at every setting of
the coil control box. Since increasing R seems to help performance, we conclude that we are still
limited by voltage noise from our current source. We generally use R = 5 kΩ, where the noise is
lowest.
194
DC Gain AC Gain R(out)current noise
(nA)field noise (fT)
1 1 50 14.000 210.000
1 1 500 1.400 21.000
1 1 1000 0.700 10.500
1 1 5000 0.230 3.450
1 0.1 50 13.000 195.000
1 0.1 500 1.400 21.000
1 0.1 1000 0.700 10.500
1 0.1 5000 0.250 3.750
1 0.01 50 15.000 225.000
1 0.01 500 1.500 22.500
1 0.01 1000 0.700 10.500
1 0.01 5000 0.260 3.900
0.1 1 50 5.000 75.000
0.1 1 500 0.600 9.000
0.1 1 1000 0.380 5.700
0.1 1 5000 0.260 3.900
0.1 0.1 50 3.800 57.000
0.1 0.1 500 0.420 6.300
0.1 0.1 1000 0.220 3.300
0.1 0.1 5000 0.043 0.645
0.1 0.01 50 2.400 36.000
0.1 0.01 500 0.400 6.000
0.1 0.01 1000 0.190 2.850
0.1 0.01 5000 0.060 0.900
0.01 1 50 2.800 42.000
0.01 1 500 0.380 5.700
0.01 1 1000 0.200 3.000
0.01 1 5000 0.062 0.930
0.01 0.1 50 2.100 31.500
0.01 0.1 500 0.360 5.400
0.01 0.1 1000 0.230 3.450
0.01 0.1 5000 0.063 0.945
0.01 0.01 50 3.200 48.000
0.01 0.01 500 0.500 7.500
0.01 0.01 1000 0.230 3.450
0.01 0.01 5000 0.069 1.035
Figure E.4: Noise measurements for coil driving circuit settings, measured using current preamplifier.
195
Appendix F: Fetal MCG report
This appendix includes a report of the type normally generated for a fMCG study using SQUIDs.
The first two attached pages are analysis of fMCG timing intervals for two different portions of
our data. The final page is from the reference data taken by a SQUID immediately prior to the
SERF measurement. Additional fMCG studies are needed to verify the consistent accuracy of tim-
ing measurements made by SERFs vs SQUIDs, but in this initial study the agreement is relatively
good.
Table F.1 shows an analysis made by Ron Wakai’s biomagnetism group of the average fMCG
waveform from figure F.1–figure F.3. The average waveform can be used in certain diagnostic
applications, such as long-QT syndrome which is thought to be associated with sudden infant
death syndrome [Cuneo et al., 2003, Zhao et al., 2006]. All intervals measured with the SERF
and SQUID systems match to within 15%, with most better than 5%. For this application, the
relevant measurement would be the QT or QTc interval. While more measurements are necessary,
comparison with the data from Zhao et al. [2006, figures 2 and 3] leads us to feel cautiously
optimistic that our magnetometer could be suitable for this kind of measurement.
Table F.1: Fetal MCG report generated for the first fMCG measurements made using a SERF magnetometerof a 31-week pregnancy. The two SERF measurements represent different measurement periods during thesame visit. The analysis was performed by Ron Wakai’s medical physics group.
Interval (ms)Method RR PR P QRS QT QTc
SERF (# 1) 425 95 42 47 264 405SERF (# 2) 426 95 48 49 282 432
SQUID 415 90 41 48 241 375
196
Figure F.1: Average SERF fMCG waveform, showing relevant intervals.
197
Figure F.2: Average SERF fMCG waveform, showing relevant intervals for a different time segment than
in figure F.1.
198
Figure F.3: 21-channel SQUID measurement for the same fMCG subject as in figure F.1 and figure F.2
199
Appendix G: Magnetic field coils
Several sets of coils were produced and characterized for this work. Where possible, the coils
were calibrated with low-frequency (10 Hz) currents applied to the coils, and the resulting response
measured by a calibrated fluxgate placed at the center. For the individual set of coils wrapped
around each magnetometer, a fluxgate could not be placed in the center, so the coils were calibrated
by centering the magnetometer inside a larger Helmholtz set that had been calibrated using the
fluxgate method. Perturbations were induced with the Helmholtz set, and the current necessary
to cancel those perturbations with the small coils was measured. Below, we describe some of the
coils details and calibrations.
G.1 Individual coils for each magnetometer
Each of the four magnetometer channels is wrapped with a single set of tri-axial coils. Each axis
is composed of two single turn loops wrapped on either side of the thermal insulation and centered
around the cell windows. Square turns were used to maximize field uniformity Kirschvink [1992].
All coils have approximate size a = 4 cm and a pairs for an axis are all separated by d = a. The
coil calibration and expected field uniformity are shown in figure G.1.
200
Cell # Coil axis Calibration µT/A
x 19.41 y 13.0
z uncalibrated
x 16.92 y 18.1
z uncalibrated
x 23.03 y 16.9
z uncalibrated
x 22.54 y 15.0
z uncalibrated
Figure G.1: (Table-left) Calibration factors for the individual coils wrapped around the four magnetometersused throughout this work. The z-coils were not calibrated. (Figure-right) A simulation of 4 cm side, 4 cmseparated single-turn coils. The contour lines represent field uniformity levels. The central square is thecell.
201
G.2 Large single coil
The large single gradiometer coil consists of 1-turn in a rectangular geometry. The coil calibra-
tion is 3 µT/A in the center. The magnetometers are offset from the center in a square pattern, and
with an offset value of 3.5 cm for all the measurements made here. Figure G.2 shows a contour
plot of the coils and the field uniformity in the center. The black squares show the dimensions of
the four cells.
202
Figure G.2: Gradient coil contour map. A single coil is in the plane of this image. The contour lines repre-sent 2% variation in the field amplitude. If the magnetometers are perfectly centered around the center of thecoils, then the fields are the same. However, any offset will impact the average field of the magnetometersat the percent level.
203
G.3 Transverse pumping magnetometer coils
For technical reasons, it is useful to use separate triaxial DC coils and a pulse coil for the
transverse pumping magnetometer. For the directions along the pump and probe laser, we use
single-turn square coils wrapped around the thermal insulation. For z direction, large DC fields are
needed, as well as a very short, high amplitude pulse.
Using the notation in section 6.4, the transverse spin-resonance is not as sensitive to the pulse
amplitude Ω1, so the field uniformity requirements are not as important for the pulse-coil. We use
multi-turn Helmholtz configuration for this set, centered around the center of the cell.
In contrast, Ωz needs to be very uniform. The resonance condition is that Ωz/q = ω1. If we
suppose a magnetic field gradient proportional to the applied field, δΩz/Ωz = c, then atoms in
different parts of the cell have different resonance frequencies, and δΩz/Ωz = δω1/ω1. If the
resonance linewidth is not to be affected by the magnetic field gradient, the resonance linewidth
must be large compared to the field inhomogeneity, δω1 Γ′. At 125 C, Γ ≈ 180 1/s (including
ΓD). Then we require cω1 2Γ, or c 0.01. In other words, we need the DC field Ωz to be
uniform across the cell volume to much better than 1%. This fact should reinforce the very narrow
resonance width of the transverse spin.
Inspection of figure G.1 immediately makes it obvious the single square set of coils will not
be adequate. Various symmetric planar geometries are available Merritt et al. [1983]. We use a
four coil system with integral turn ratios. The geometry can be seen in figure G.3. Integrating the
average deviation from Bz(0, 0, 0) along the probe beam gives a linewidth of less than 2 Hz, and
the deviation in the whole cell should be smaller than 0.1%.
204
Figure G.3: The coils used in the transversely pumped magnetometer to produce very uniform Ωz . Theuniformity is better than 0.1% in the cell volume from these coils, which should not broaden the transversespin resonance until the resonance frequency is several MHz.
205
Table G.1: Calibration for all axis of the transversely pumped magnetometer. The numbers below the coilcolumns are the calibrations measured in the coil centers by a fluxgate. The matrix represents measurementsof how orthogonal coil axis are to one another, and are normalized by the total calibration in the top column.
Bpulse Bx By Bz
242 µT/A 30.6 µT/A 29.8 µT/A 251.7 µT/A
Bx 0.03 1 0.04 0.003By 0.03 0.01 1 0.05Bz 1 0.01 0.03 1
206
Appendix H: Magnetometer parts list
Table H.1: A list of the custom parts used in the magnetometers, both designed as part of this work andpurchased from outside vendors. Custom parts which are adapters for external optics have the optic part #listed.
Description Vendor (adapted from)
1 in adapter for 18 mm large-diameter as-pheric lens
custom (Thorlabs)
SM05 adapter for 11 mm aspheric lens custom (Thorlabs)SM05 adapter for 8 mm aspheric lens custom (Thorlabs)SM05 adapter for 4.51 mm aspheric lens custom (Thorlabs)SM05–SM1 adapter to mount smaller lensesin SM1 tubes
custom (Thorlabs)
SM1 retaining ring custom (Thorlabs)SM1 FC-APC fiber connector plate custom (Thorlabs)SM05 FC-APC fiber connector plate custom (Thorlabs)SM05 retaining ring custom (Thorlabs)Two-axis mirror mounts custom (for prism mirrors)Tri-bore optics tube customCeramic heat-sink customMagnetometer plastic scaffold customphotodiode adapter piece customHeater-controllers Custom (see appendix E)Coil drivers Custom using circuit in appendix E
1 in Glan-Taylor polarizer, AR-coated795 nm
United Crystals
1 in 10 cm clear aperture 0-order waveplateat 795 nm
United Crystals
0.5 in Wollaston beam splitter United Crystals0.5 in 0-order half-waveplate at 780 nm United Crystals0.5 in Glan-Taylor polarizer United Crystals
Lasers (Pump)Eagleyard Photonics EYP-DFB-0795-00040-1500-BFY02-0002 (80 mW)
Lasers (Probe for transverse magnetometer) EYP-DFB-0780-00080-1500-TOC03-0002
Lasers (Probe for array)Sacher Lion Littman Metcalf ECDL(25 mW)
Lenses (diffusion probe) f=25 mm Thorlabs #AC127-025-Bf=-15 mm Thorlabs #LC2265-B
207
Table H.1: A list of the custom parts used in the magnetometers, both designed as part of this work andpurchased from outside vendors. Custom parts which are adapters for external optics have the optic part #listed.
Description Vendor (adapted from)
f=30 mm spherical singletLenses (DC-probe) f=11 mm aspheric lensLenses (diffusion pump) f=4.51 mm aspheric lens
Lenses (DC-pump)f=18 mm large aspheric lens Thorlabs#AL2018-B
Polarizer (thin) 0.5” Thorlabs #LPVIS050
MirrorsDielectric prism mirrors Thorlabs #MRA10-E03
Fiber mating sleeves Thorlabs #ADAFC2-PMNLaser controller (current) Thorlabs #LDC202CLaser controller (temp.) Thorlabs #TED200C
FibersOz-optics 6 m polarization maintaining#PMJ-3A3A-633-4/125-3-6-1
Fiber splitters5-way (4-equal power, 1 5%) Oz-optics#FOBS-15P-111111-5/125-PPPPPP-780-95/5,50/50-40-3A3A3A3A3A3A-3-1
Aerogel insulation from McMaster-Carr part no. 9590K8
HeatersMinco thermofoil heaters,HR5578R4.6L12A (ordered from dis-tributor, Temflex controls inc.
ThermistorsNTC Thermistor 112-103FAJ-B01, fromNewark
Data Acquisition equipment National Instruments FPGA DAQ system